I have inherited a monster.
It is masquerading as a .NET 1.1 application processes text files that conform to Healthcare Claim Payment (ANSI 835) standards, but it\'
I would start with uninhibited use of Extract Method. If you don't have it in your current Visual Studio IDE, you can either get a 3rd-party addin, or load your project in a newer VS. (It'll try to upgrade your project, but you will carefully ignore those changes instead of checking them in.)
You said that you have code indented 15 levels. Start about 1/2-way out, and Extract Method. If you can come up with a good name, use it, but if you can't, extract anyway. Split in half again. You're not going for the ideal structure here; you're trying to break the code in to pieces that will fit in your brain. My brain is not very big, so I'd keep breaking & breaking until it doesn't hurt any more.
As you go, look for any new long methods that seem to be different than the rest; make these in to new classes. Just use a simple class that has only one method for now. Heck, making the method static is fine. Not because you think they're good classes, but because you are so desperate for some organization.
Check in often as you go, so you can checkpoint your work, understand the history later, be ready to do some "real work" without needing to merge, and save your teammates the hassle of hard merging.
Eventually you'll need to go back and make sure the method names are good, that the set of methods you've created make sense, clean up the new classes, etc.
If you have a highly reliable Extract Method tool, you can get away without good automated tests. (I'd trust VS in this, for example.) Otherwise, make sure you're not breaking things, or you'll end up worse than you started: with a program that doesn't work at all.
A pairing partner would be helpful here.