It has been raing a lot in los angeles and because of that water has been leaking into the water heater room, which is located in the upstairs balcony. Water has started collecting in the kitchen ceiling. How do I fix it and would I be a good or bad idea to make a hole in the paint where the water is leaking from to drain it?

Might as well make that hole to help the house drain and start to dry out. Better yet get on the roof and fix the leak so your new drain hole won't just kep draining and draining and draining. Is there a chimney or vent cao above this room, check there first for a leak in the shingles or flashing