Try looking here: http://support.apple.com/kb/HT1431#faq8
Resetting the PMU might do the trick. I know the ol' Titanium is sweet, but it also might be time to look at a newer model.
One thing to check would also be the power cable from the adapter to the computer. If there's a physical break in that line it would result in the same issue, slow battery decline and failure of power. Find another AC adapter and see if it works. If that still fails, then Tyler's advice stands.
How on earth do I get the black ribbon cable back into its connector???
Cor, You have to slide the little bar on the connector away from the body of the connector before you can insert the ribbon cable into it. I know it's tight quarters, but even my meat hooks were able to finally get it inserted. The 'trick' is that once you have inserted the cable into the body of the connector, quickly slide the bar back toward the body as this will lock the cable into the connector. I hope this is helpful.
I replaced the battery in both my iPhone and my wife's in the same session. Using the slideshow and directions I was able to successfully do the replacement. Of course my iPhone was the first one done so as to gain some experience before tackling my wife's iPhone. I had two minor issues in reassembling mine. Firstly, I completely missed re-installing the number 3 cable. It wasn't till I was reassembling my wife's iPhone that I realized I'd missed it. Not sure what it supports as I had my phone up and running, checked the GPS, etc while it wasn't connected. Secondly, my camera wasn't seated properly and I got the strangest imagery when I launched it. All green and black, looking like one of those ghost buster reality shows using night vision. Fortunately, both issues were easy to correct. Outstanding directions and slides! Oh yeah finally, the kits for the battery comes with both a #00 Phillips screwdriver and a spudge, however, I ordered the extra spudge and got one that is better suited for prying the battery up.