I honestly don't even see how a poorly done tile job can leak this quickly. I can tile and grout a wall pretty sloppy and it will be good for at least 4-6 month and possibly a lot lot longer.
The short-term fixes you can do now are exactly that at best - short-term. It is possible that you will spend money trying to waterproof after - which in no way is ever the answer - and it could possibly have no impact.
You need to get your money back from this contractor. Even a bad contractor puts up some sort of water barrier in a shower. A bad contractor would put up the tile with a bad grout job or wrong grout and maybe the drywall gets wet and eventually moldy. Your contractor either has no clue or is insanely lazy. You cannot let him back in your house.
(Not going to be as helpful as I'd hoped, but maybe helps you think of something you hadn't thought of)
Since you mentioned the kitchen is right below, I'm going to assume finished walls/ceilings and that you can't easily trace the pipes(or maybe you're lucky and it's a drop-tile ceiling?).
I've heard of a few ways to find a leak, but they almost all involve being able to reach the pipes - either to spray something like dish soap on them and watch for bubbles or put an improvised stethoscope(flat-head screwdriver, usually) to them so you can listen for the leak.
I've had both tub/toilet gasket go and not be obvious for a few days (wood subfloor wicked the water to about 4ft away in both cases), so that might be something to consider too.
Either way, there's a pretty good list of steps to try here:
https://www.plumbingsupply.com/how-to-check-for-leaks.html
Most of it depends on having access to the piping though.
Having said that, unless the local plumber has a camera s/he can fit down the copper pipe (?), they're going to have to make holes too...
Best Answer
You have a roof leak that is dripping either directly onto the fan assembly or onto or into the vent tubing. Probably the leak is around a roof penetration, but we had one due to a defect in the roof decking right above the fan. Water dripped onto the fan assembly, through it, and into the bathroom.
EDIT Another possibility is condensation of water in the tubing connected to the outlet of the fan. The tubing is in the cold attic and so water vapor will condense on the walls of the tubing conducting the exhausted air to the outside. If there is a lot of condensate in the tubing when the fan is turned off, and if the routing of the tubing is level and upward then this would drain down into the fan assembly unless there is a drain to catch it and conduct it away.