(Note: the first two paragraphs of this answer were aimed at the original title of this question, which was "Why does the ball no longer point straight down in a multi-engine airplane with inoperable engine?")
First let's address a misconception that some readers may take from your title. In linear, constant-heading flight in any aircraft, "coordinated" or otherwise, the slip-skid ball always "points" straight down in relation to the earth. If you have a conception that in linear, constant-heading flight in a multi-engine aircraft with a failed engine, the ball is not pointing "straight down" toward the earth, then you have a misconception-- regardless of whether or not we are applying the optimal rudder input and bank angle to keep drag to a minimum.
But "straight down" is not the same as "centered", if the aircraft is banked. More to the point, regardless of bank angle, and regardless of whether the intended flight path is a straight line or a turn, it turns out to be not optimal to keep the slip-skid ball exactly centered, when dealing with a failed engine on a multi-engine aircraft.
Now moving on to your actual intended question as I understand it-- I'm assuming that the real essence of your question is "why is not optimal to keep the slip-skid ball centered, in the case of engine failure in a multi-engine aircraft?" Or to put it another way, in this case, "why is the ball not centered in 'coordinated' flight?"
You need to start by asking yourself what is the meaning of "coordinated flight". We could define "coordinated" in several different ways, one of which would be that "the ball is centered". But if by "coordinated" we mean that the the fuselage is pointing straight into the relative wind, not yawed to one side-- i.e. if we mean that a tuft of yarn ("yaw string") mounted in the middle of the windscreen would be centered, not blowing to one side-- then we don't always want the ball to be fully centered, especially in the case of a failed engine on a multi-engine aircraft.
What's going on here is that when the fuselage is fully streamlined to the relative wind and not yawed to expose either side of the fuselage to the airflow-- the optimum situation for minimizing drag-- then the ball cannot be fully centered, because the rudder itself, which is strongly deflected to compensate for the failed engine, is generating a significant aerodynamic sideforce, and therefore a significant acceleration, toward the failed engine. Compensating for this sideforce by banking slightly toward the good engine stops the flight path from curving, i.e. stops the aircraft from turning, so that the aircraft's heading remains constant, but doesn't affect the position of the ball, so the ball continues to be off-center.
If we applied enough rudder pressure to fully center the ball, this would mean that we were exposing the side of the fuselage (the side nearest the failed engine) to the airflow, thus creating an aerodynamic sideforce that exactly cancels the sideforce from the deflected rudder. (And in this case we could hold heading with the wings level-- there would be no turning tendency.) This is not optimal--drag is greater than it needs to be.
Note that in both cases the torque from the deflected rudder is cancelling the torque from the thrust asymmetry. In the latter case, the slightly greater rudder deflection is actually creating a little extra torque, that is used to keep the fuselage and vertical fin pointing slightly sideways in relation to the relative wind and airflow-- again, this is not optimal.
In theory this happens any time we deflect the rudder for any reason. If we want to keep the fuselage streamlined to the airflow, and the rudder is deflected at all, the ball should not be fully centered. Rather, we should apply slightly less rudder than would be needed to fully center the ball. Otherwise we are exposing the side of the fuselage, the side opposite the deflected rudder, to the airflow. But normally this effect is trivial enough that we don't worry about it. Not so when the rudder is strongly deflected to counteract the torque from a failed engine.
We could imagine some other system of yaw control-- such as the "clamshell" split ailerons on the B-2 Spirit-- that would create plenty of yaw torque, but no significant aerodynamic sideforce. In this case, if there is a thrust asymmetry, to fully align the centerline of the aircraft with the direction of the flight path, relative wind, and airflow, the ball should simply be centered, not left deflected to one side.1
Maybe it would help to explain it this way: one way of looking at things is to say that you should use the rudder as needed to hold the heading constant, and then, while continuing to hold the heading constant, bank toward the good engine enough to move the ball slightly off-center (one-half diameter is often recommended). But another way of looking at things is to say that you should apply rudder as needed to bring the ball nearly but not fully to center (again one-half diameter off center is often recommended), so that the fuselage is fully streamlined to the airflow, and then, while keeping the ball in that position, bank as needed toward the good engine to stop any turning tendency and hold the heading constant. In theory, you'll end up in the same place with either method. Granted, in the real world, there are reasons why it's very important to immediately counteract any yaw toward the failed engine, so the first method-- prioritizing holding the heading constant-- may be better. But thinking through how the second method would work, may give you a better understanding of what is really going on in terms of aerodynamic forces and torques.
PS-- I guess I'm starting from an assumption here that you understand the difference between sideforce and torque. I'm also starting from an assumption that you understand what the slip-skid ball (inclinometer ball) really indicates. The slip-skid ball responds to the net sum of all lateral (sideways) accelerations acting on an aircraft, excluding the lateral acceleration component due to gravity. And this is using "lateral" or "sideways" in relation to the aircraft's own reference frame-- that's why banking doesn't affect the position of the ball in and of itself, nor does it generate any aerodynamic sideforce. (Obviously it's a different story if we are also changing some other variables, for example if we are manipulating the rudder as needed to force the flight path to remain in a straight line as we vary the bank angle, in which case we'll be demonstrating a sideslip.) For (much) more, see this related ASE answer: What does the balance ball actually indicate?
See also this related ASE answer: Why should you not turn in the direction of an inoperative engine?
And see also this related answer: How can using split throttles help when landing twins in crosswinds?
Closing note: note that this answer and at least one other posted answer are basically in agreement, except that this answer uses the word "sideforce" specifically in relation to the aircraft's own reference frame. By this convention, banking creates a horizontal (sideways?) force that keeps the flight path from curving (keeps the aircraft from turning), but does not actually create a "sideforce" in the aircraft's own reference frame, hence the bank angle does not directly affect the deflection of the slip-skid ball. Also, the present answer, while possibly more verbose than absolutely necessary, is intended to encompass situations where the aircraft is not simply travelling in a straight line.
Footnotes:
- But on the other hand, minimizing sideslip (centering the yaw string) is not that important, in terms of minimizing drag, in an all-wing aircraft like the B-2, with minimal cross-section as seen in side view. Consider this quote from the 35th Wright Memorial Lecture by Jack Northop on the development of flying-wing aircraft: "for very long-range aircraft there is a valuable compensating advantage in being able to fly under conditions of asymmetrical power without appreciable increase in drag." In such a case the pilot should simply make whatever control inputs most efficiently balance the aircraft in roll, with minimum drag penalty, and so long as the aircraft has efficient ailerons, minimizing sideslip (centering the yaw string) will be less important than in a more conventional aircraft.
