Well, I am about 20.000 miles away, but I may give you an advice.
Try lifting the white-ish caps-sliders from switches on the top of the unit. Also, there is one more screw in the battery compartment on you pics. If you removed it, than keep the tape door open, and try gently again.
If it looks like the tuner string is in the way, as baddboybill said, that is going to take a lot of fiddling around, but it can be done. The string can be released from leftmost pully/part of the loop and then placed back after the tape deck is done.
Just keep the tuner string away from fire and high temperatures. That string is impregnated with wax and will burn fast, and replacing it is rather hard. For example, the soldering iron will cut through it instantly.
As far as the broken switch is concerned, you can glue it with Acrylate glue later (3M, Loctite, whatever 10-second glue), but consider it the last step in the process, after everything is checked OK, and you have placed all screws in. Do not be alarmed, we have all broken somethuing during restoration/repair. Just as H Ford said, mistake is an opportunuty. It was broken when you started, was it not?
Did you clean the battery connectors? If you can't find anything else, basic household vinegar will do miracles (it will foam a little), just take care not to splash it on circuit board, and rinse it with water. Take a cotton cloth and wipe batt compartment afterwards. The chemical principle of cleansing the oxidation is rahter simple, the oxides were made due to alkaline chemicals in batteries, and you are neutralizing them with (rather weak) acid. Not exactly the paper-mesh vulcan, but similar.
Take care!