So a got the new cap in (I finally found a place that's local that stocks this stuff!) and it's working great! Thanks to everyone who helped me out here! This is a great forum!
I'm going to note some things here for future reference so I can build on what I learned.
My Steps For Repair:
1. Test external power supplies.
2. Visually inspect components on PCB's looking for things that are blown or burned.
3. Check input and output of voltage regulators.
4. If there's a problem on a voltage rail then check each component on that rail until you find the problem.
Things I Learned:
-Desoldering braid is really annoying. I need a faster desoldering solution.
-Electrolytic capacitors tend to go bad so check those first.
It's a matter of getting used to the tools and knowing which to use for which job.
Personally (and this is personal taste, almost everyone has a different preference) I usually use braid for cleaning up after I've desoldered, I find it frustratingly long winded and awkward for most desoldering jobs (it is useful for desoldering seams or removing 'bulk' solder).
If you're going to use it, buy decent braid, there is a world of difference between a nasty cheap braid and a high quality (expensive) one from Chemtronics (Soderwick, hate the name, love the product)
I prefer a decent solder sucker, I find they work very well on through hole components, single sided, double sided and multilayer through hole plated boards.
Other people will tell you they destroy boards and are useless.
I believe it to be a skill thing, some people get on with them, others not so much, buy a foot of small bore silicone tube that's a tight fit to the sucker tip, cut off 5-10mm lengths and slip one over the end of the solder sucker tip so it extends the tip by a couple of mm, it saves the tip and, if you're not practiced in the technique, it also saves the board.
If you have problems removing solder, add solder, it's easier to suck up a lot of solder than a little and wetting the joint will make it flow better anyway.
I've also tried desoldering irons, the Pro's Kit one I have is crap, similarly the (very expensive) OK Industries one I used to have was better but generally more effort than it was worth and I found I spent far more time maintaining it than it saved.
Other people swear by them, YMMV, unfortunately the only way to find out what works for you is to try it and see.