Can you drain and solder it?

I forgot to mention that I had tried that, but I had some trouble
getting the solder to adhere. My best effort seemed a bit....lacking.

Are there special techniques for soldering brass?

Clean all surfaces, use flux and the right amount of heat. If there is any
water in the line it's not going to work.

I think this will be a hack though. Your best bet is to take it all apart
and fix it right. Many times I've spent more time trying to make something
work than to simply do it right in the first place.