Lights stopped working
It takes very little power to get an LED light to glow. In addition to strange glowing like you have found, this is why some LED lights will glow or flash with certain "smart" switches or timers or other special types of switches even when they are turned off. So I wouldn't worry about the glowing when you touch them.

The real question is why they are not working any more. There could be many reasons, including:


Circuit breaker tripped - that is easy to check and reset.
GFCI/AFCI tripped (which can be on a circuit breaker or elsewhere in the circuit) - that is also easy to check and reset.
Bad connection somewhere in the circuit (breaker, switch, light)
Lights "burned out" - this is actually least likely because it is unlikely that both lights would "burn out" at the same time, though possible with a big surge.


If the lights are removable "bulbs" then it is reasonable to remove the old ones and try replacements - just make sure they are the same size/type/power rating. However, many LED lights are integrated fixtures, so that may not be an option.

So check any circuit breakers and GFCI/AFCI test/reset buttons first. If resetting a breaker or GFCI/AFCI solves the problem and the breaker or GFCI/AFCI doesn't trip again then you are set. However, if you reset a breaker or GFCI/AFCI and it trips again when you turn on the lights (or turn on some other device on the circuit) then you have a serious problem that needs to be investigated.

Actually, while a single GFCI or AFCI trip can be due to lots of "no big deal" reasons, even a single regular breaker trip (i.e., overcurrent) can be a bit of a concern if there is no obvious cause.

In any case, since this is a rental, if a simple reset does not solve the problem then you really need to either have your landlord hire a professional electrician to troubleshoot the problem or hire one yourself. You can't do electrical work in a rental yourself in most locations and for a good reason: If you mess up and you are not a professional electrician then your landlord will claim it is your fault, you will claim it is was faulty wiring in the first place (the landlord's fault) and the insurance company won't pay.