I don't have a 770 but I have had some experience like this recently...
If the suggestion that tfflivemb2 provided does not help, you may have to obtain a digital multimeter to further isolate the problem. Doing so will allow you to measure the voltage on the main battery, the voltage being supplied to the battery from the terminals in the laptop and then, if my suspicion is correct, test a fuse on the motherboard which may have blown. You can get a fairly cheap multimeter from Radio Shack or Home Depot. Something that measures AC and DC voltages up to 200v (good enough for your laptop and around-the-house usage) and resistance. Auto-ranging is nice to have but adds to the cost. Also, download the Hardware Maintenance Manual (HMM) from here if you don't already have it:
ftp://ftp.software.ibm.com/pc/pccbbs/mo ... 7l5152.pdf
Referring to page 37 of the HMM, you should be able to get the leads of a multimeter into the slots of the battery to measure the voltage. If it reads close to the rated voltage, your battery is probably okay. (I'll ignore the current capacity for the moment). Power up the laptop with the AC adapter and
carefully put the leads on the two terminals inside the battery compartment. Looking at the HMM, you may have to do this with the laptop partially disassembled. If you don't get any voltage at the terminals, or you feel that you cannot measure it safely, then you should look for a fuse and test it.
I'm assuming for the moment that IBM has designed their laptops in a similar manner over the years. I have a T23 which would not power on when run from a known good AC adapter, yet worked fine on battery power alone. The problem was a blown 7A fuse on the motherboard. In fact, there appear to be separate fuses protecting the DC power input, the main battery bay and the Ultrabay. Your situation may be like mine. If there is a blown fuse in-line with the main battery bay, it will not charge nor power the laptop. Look around the area where the battery terminals are. You can usually identify fuses as being labelled as "F7" or some such number. With the laptop completely disconnected from any power source (AC and battery), put the multimeter in the lowest resistance range and place the leads across the ends of the fuse. You should read zero ohms if the fuse is good and "nothing" if it is bad.
If you
do find a bad fuse, your problem would normally just be starting. You now have to identify the amperage rating
and find a source for the replacement in single-unit quantities. There may be markings on the fuse which can identify the amp rating and/or a manufacturer code. If you can't find anything, PM me and I'll see what I can do to figure it what it is. For an answer to the second problem, I'll pass on this piece of advice: I have used both DigiKey.com and Mouser.com as a source of parts since they both usually offer to sell just one unit with no minimum dollar limit.