It doesn't look so bad (you just need to clean up the holes from the pictures that you posted), so I would
a) clean it up a bit with some alcohol, using a Q-tip or similar appliance (don't get heat near the alcohol, you don't want to catch it or yourself on fire)
b) get a pair of locking pliers and a needle that has the same diameter (close is fine) as the lead on the capacitor
c) put the needle into the hole, while holding it with the locking pliers, then heat the needle close to, but not touching the pad/hole and pushing slightly, so that it melts the old solder and pushes it out of the hole.
d) clean it again with alcohol, using a Q-tip or similar appliance (see previous fire warning)
e) (my personal take - get rid of that crappy lead free solder, use some nice 85/14/1 with flux) and an appropriate (not high) heat. Test by applying the solder to the tip of the soldering tool, if it melts quickly without scorching, you are ready to rock-n-roll
f) insert the capacitor (correctly oriented) into the newly clean holes, heat the wire attached to the capacitor near the holes and apply new solder, allowing it to wick onto the pad/hole/wire assembly.
Do not use a hand drill, unless you have done that before and KNOW what you are doing.
You can substitute IPA for alcohol if you got it.