There's not much else they can do in a situation like this. ATC, in this case the tower controller is the only person(instance) that has a comprehensive enough view (both literally and metaphorically) of the situation to give orders, clearances etc.
What the pilots can and will do in a case like this, is proceed according to their last clearance. If you are cleared to land, you land. If you were cleared to taxi somewhere, you taxi untill you reach the clearance limit (typically a holding point, gate, or place at apron) and so on.
When you reach the clearance limit, if you want to proceed you ask for further advice/clearance. After landing you would vacate the runway, then ask for taxi instructions (if none were automatically given). If you were taxiing for takeoff, and reach, say, a holdingpoint, you'd tell ATC you are ready for takeoff at that holdingpoint. If you are on approach, but not cleared for landing, you obviously have no other option but to conduct a missed aproach as per published procedure.
When you reach your clearance limit and you don't hear anything from ATC, or as in the video, the communication is incomprehensible, you simply stop and wait (or go around) and keep trying to communicate with ATC. And as in the video, if this situation persists, you try establish communications with someone else who can check the status of the ATC.
One should keep in mind that despite having an ATC control your movements, as a pilot you still have to maintain an alert lookout. You allways have to be aware of what is happening around you, by looking out, listening to radio communications etc. Someone might have misheard/-understood their clearance or act unexpectedly for any other reason, and you have to be "awake" and ready to adjust your actions.
*Edit: I'll take the liberty to add some hindsight here. From the very beginning of the recording it is dead obvious there is something wrong with the controller. Even without seeing her, it's pretty clear she's either under the influence, or having a stroke, latter of which she was. It's kind of disturbing that none of the flightcrews either recognized this, or if they did, decided to let it go (at least one obviously did notice the gibberishly speech and made a remark about comedy or something). It's also weird that the controller kept on working throughout the stroke, but I'm guessing the stroke had impaired her judgement.
So: if you ever encounter someone speaking like that, check out what it's about. You might end up saving that someone's life.*