Some of your assumptions about the wiring are probably incorrect.
There are two ways to hook up a light switch. One is that you bring the circuit up from the branch feed to the switch, then switch the hot, and continue both the switched hot and the neutral up to the fixture. The second way is that you bring the hot and neutral up to the fixture, but then you take a pair of wires down to the switch. The incoming feed at the fixture is always hot, and is connected to one wire going down to the switch. The wire coming back from the switch is the switched hot. If you do this with typical white and black wires, you SHOULD wrap the end of the white wires with black electrical tape to indicate that it’s really being used as a hot wire and not a neutral. In practice, I almost never see the colored tape.
The second way is very common for fans since it results in less wire being used. You don’t have two separate feeds (one for the fan and one for the light). You just have one feed and the wire necessary to go down to the switch.
Your LED transformers should be connected in parallel (black to black, white to white). You shouldn’t have them on two separate feeds like your picture shows. You need to figure out how your wiring is connected, and without the proper tools I don’t know how you are going to figure that out except for random guessing, and that’s a good way to burn your house down. If you don’t want to spring for an electrician, at the very least go down to ye ol local hardware store and buy yourself a cheap voltmeter. At this point though, I’d really recommend an electrician. You seem to be in way over your head.