Repairing a circuit when the break is unknown and inaccesible.
One possible investigatory tool is a "toner" set - Fox &amp; Hound or some other brand - While they are are more often used on telecommunications cables, some do claim to work on power wires. User reports vary, possibly due to different toner sets working more successfully than others with power wires. They work by one part applying a small electrical signal to the wires, which  the other part of the set can pick up without having to actually touch the wires. If you can get a tone on a circuit (Disconnect it at the breaker before hooking up the toner - assuming you understand the risks of and precautions to take when working in the breaker panel [scratch that - it's the landlord's problem - leave it alone]) and then you find a place in the wall where tone stops, rather than going off in any of the possible directions at that point in the wall, it might mean you've found the break. Check both sides of the wall, etc. to increase the odds of success. Try connecting the toner at one of the dead outlets and tracing backwards, as well. If they both stop in the same place coming from different directions, that's a good place to knock a hole in the wall.

Before any more fail, map whatever circuits still work, and perhaps turn off any you can't find out what's connected to (paying attention to things that might not be instantly obvious like furnace or boiler, well pump if any, sump pumps, etc...)