There could be a short that I can't see, but the "other end" puts the batteries in series.
By "convention" positive is switched, but it doesn't matter.
This you need a lead to a free battery terminal - (likey negative)
If you temporarily attach the other wire to the positive terminal, the toy should work without the switch. If it doesn't with fresh batteries, there is no point in going further. You also can't EXPECT reverse polarity protection.
A slide switch has 3 terminals e.g. (1) (2) (3)
(2) is the common terminal
1, is connected to 2 when the switch is pushed toward 1
3, is connected to 2 when the switch is pushed toward 3
Thus, the middle terminal and one outer one will get the job done. if there is an ON/OFF marking, then that could be reversed..