broken left-hand hinge between body and screen

Unfortunately the hinge on a Titanium is part of the screen assembly, and it's not possible to swap out a left hinge or a right hinge -- you'll need to replace the whole display assembly, screen, hinge, cabling, and all. Technically it is possible to transplant your screen into another casing/hinge/bezel, however I've found that removing the bezel from a Titanium effectively destroys it, so replacing the whole thing is really the only way to do it without pulling your hair out in the process.

I'd recommend buying a dead Titanium on eBay that has a good screen and hinge. Just make sure to buy one from a machine that has the same model number and processor speed, because there are at least a couple versions out there and it would be a bummer to end up with one that has the wrong video cable.

Another note -- if you current screen is working, you can make up for the cost of the above purchase by stripping off the bezel/casing/hinge and selling the screen itself on eBay as a working bare screen. There are a number of 15" laptops out there which use that screen, and, who knows, you might end up making money at the end of the day.

Also, taking apart a Titanium to the degree necessary for screen replacement is one of the more difficult procedures out there, and many who have taken it on have sworn to never do it again. I'm not trying to discourage you, but if it's not too bad, you may want to consider just putting up with it.