That is one old computer, very nice, i can see the eisa slots(or is it vesa?), but unfortunately the pictures 1 and 3 are too blurry to be see what is in there.
By ps2, are you talking about the port to connect the mouse yes? If so, I don't know if the wire in the picture 1 is the right one, on those old machines what was used to connect the ps2 and USB used to be a bracket adapter like this one
(those were very common around 1995-2000) or maybe this one
is what you have, in any case you need the motherboard manual to learn where to connect your adapter.
Could you please post the motherboard model and lets figure out how yours work, post too the other hardware aspects like processor, RAM, video etc, you got one old working machine there which is very cool.
By the way I don't think you need to solder anything.