No, my soldering looks like garbage 
.

I have been there....yeah it is garbage, but the garbage soldering gets better with practice.

I personally have not hacked a gamepad, so I might end up corrected here. Each of the circles you soldiered on is, as javeryh pointed out, two sides of the connection. When you push down on a gamepad button, it puts a conductive material over the both halves of the circle and completes the circuit.
You will need to solder on wire to either side of the connection on the circle. Then each of those wires to either lead on the microswitch. I can't tell if that gamepad has a common ground, so you may need to solder 2 wires for each button, so 4 wires total.
My advice is to first remove all solder from your gamepad...then try soldiering a wire to each of the "mickey mouse ears" on the circle. Be careful to cross over to the other half. (Edit: I mean remove the solder you added, not
all solder).