During a control tower malfunction, can planes still land?
If one tower goes down, the other layers of ATC will still be functioning, and they'll divert the aircraft elsewhere: if LAX goes down, national level ATC would work with the SoCal Tracon (Southern California Terminal Radar Approach Control -- i.e. "Approach") and perhaps the airline dispatchers to send the airplanes elsewhere: San Diego, Ontario, Long Beach, Burbank. Those farther out would be diverted elsewhere just to avoid having too many airplanes piled up in one place: land in the Bay Area, or LAS or PHX, for instance.
Okay, let's assume that all the towers in the LA Basin are all down. And let's assume that SoCal is down too. And just to make the scenario complete, we'll assume that LA Center has lost all of their radios too. (This is like the scenario that happened several months ago when Chicago Center lost their radios due to sabotage, except we're assuming LOTS of facilities all have the same problem at the same time. Straining the limits of "willing suspension of disbelief" pretty hard here, but let's do it to make the point.) What happens?
The pilots talk to each other on the usual frequencies and land.
There are plenty of uncontrolled airports out there, and they function just fine with pilots using what's known as CTAF -- the Common Traffic Advisory Frequency, where everybody broadcasts their position & intentions. "Whozit Traffic, Cessna 1234A, left base, touch & go Whozit." "Whozit Traffic, Piper 9876B, left downwind, full stop Whozit." Based on those transmissions, you can build a good picture of who's where, who's closest to me & where I need to be looking to find him, and what (if anything) I need to do differently to not get too close.
And nobody needs ANY assistance from the tower in order to put the aircraft safely on the runway, stop, and exit. We talk to them to keep the flow of traffic neat & orderly (they can be more efficient than just working it out on our own -- particularly when you start alternating departures with arrivals), but they have zero role in the actual act of landing.
Now in the scenario posited above where every ATC radio in all of the southern half of the state of California has failed, there would be complications, particularly if there was weather involved. (See Die Hard 2, where weather had shut down EVERY airport in North America, apparently, save one, and the bad guys had shut down the ILS to that airport... we're well beyond "willing suspension of disbelief" there.) But you'd also have a couple of things in your favor: no departures, and the runways normally used FOR departures would be available for landings as well. Also, the goal is no longer a smooth & efficient flow of aircraft in & out; it's simply to get everybody on the ground safely.
Would it be a mess once you got on the ground & wanted to get to a gate? Sure. Visions of gridlocked LAX are easy enough to imagine. But could everybody get on the ground safely? Absolutely.