I have a pretty large budget and am extremely motivated. But what servos would I need? And do you think i would have to build two arms to handle the wiring, solder wire, and solder tip?
You should start with planning, making sketches of different ideas and then put them aside for a week or two (while doing other parts of the project). The subconscious is a great problem cruncher and when you go back to the sketches, you will have reached some conclusions, seemingly without any work.
Then make simple prototypes of the best ideas, but keep it simple, you shouldn't build the entire thing, just sub-assemblies, to make sure that it will work as intended - sometimes a cardboard prototype will do.
Like your mention of a 2 DOF arm to control the soldering iron...
If you made even the simplest model (in cardboard, Mechano, LEGO or whatever), you would realize that 2 DOF isn't really gonna do it - you need to be able to move it in 3 planes to position it (X and Y) and lower it to the joint (Z).
And did I mention I have never built a robot before, have no idea how to program, and have only a rudimentary understanding of electronics.
Now this statement makes the rate of success somewhere between slim and none and my best advice i to buy either a finished robot meant for this app. or hire a skilled engineer to do it.
Had you written something like: "I need it to solder a wire with a diameter of 0.02 mm (tol. 0.001 mm) to a pad measuring 1 mm +/-0.03 mm..." etc. I would have determined your chance of success a fair bit higher, as this is how engineers approach a problem.
So, for starters, throw some numbers around and think in modules: One "arm" to feed the wire, on arm to place the soldering iron, a third "arm" to apply solder and when each is doing a proper job, you can build them into a combined robot.
For detecting the position of really thin wire, a linear opto-array is the way to go - they're used to detect irregularities in thin tread in industrial weaving machines among others and they can be had in resolutions in the few Ám range.
The same array could be used for positioning it precisely over the part, used as feedback in the servo circuit you're gonna need (don't confuse this with R/C servos, they wouldn't be anywhere near the precision you need).
You need something to aim the soldering iron as well. A LASER pointing to the pad could be used for easier aiming, if you want to move the iron around, but as someone mentioned, an X-Y plane moving the board around under a soldering iron that just have to be lowered.
You also need to figure way to clean the soldering iron in between each soldering.
The best "robot" for such tasks is still young girls (probably since they'll work at way below minimum wages in some contries, just to keep hunger at bay) - even the dumbest person will be way better than anything you can build in a year with such a modest budget - $200,000 is a more realistic budget for such a project, at least if you want a (western) company to make it.
Building robots, you need to be skilled in mechanical and electronic design, as well as programming - remove one and the project won't be done the way you intended it.
Just took a second glance at your photo and it still doesn't tell me what it is that you want the robot to solder?