'91 honda accord battery light on with new battery.

http://www.manualslib.com/manual/464695/Honda-Accord.html here is 90 91 92 93 honda accord manual

Questions:
Is it a carburetor or fuel injected model?
Did the slow take off occur before or after you changed the plugs, wires, and timing belt?

If it started after plugs, wires, and timing belt, I tend to believe the timing relationship between the cam and crank has changed (cam and crank are not timed together properly, off by a tooth or two). Unfortunately you will need to partly tear into it again to verify the timing marks used when the timing belt is changed are perfectly aligned.

I have seen issues like this on other Honda's and the problem turned out to be the ignition starter switch. Do this, jumper power directly to the starter solenoid trigger wire and see if the engine cranks.

With the engine running are you getting any voltage off the alternator? check with a multimeter. Check voltage at battery while running should be 13 volts or more.

I would suggest looking at the little things. Check your fuel filter, and clean your spark plugs. No Guarantee that this will work, but this is much cheaper than replacing something expensive that you don't need.

OMG!!! I have the same problem with my 09 accord. Mine has died 3 times as well...and the breaks are sooooo loud. My friend has a 09 civic and her battery dies as well. Honda needs to fix this problem.. I regret buying a HONDA!!!! Dont buy one.

Either your alternator is not charging properly or something is draining your battery while the car is off

If you manage to get it started, check the alternator output...I'm betting it is bad. Should be making no less than 12.5 with accessories all on, 13.5 to 14 with just engine running. Bad alt can drain battery even when off depending what failed inside it. Towing with a chain is not really a good idea. the cross pin in those differentials are not very good and can fail.

sounds like a bad battery or the battery cables are corroded where they hook up to the battery or loose. if it cranks for a couple seconds slow than clicks quickly there isnt enough voltage getting to the starter. not all new parts are good.

It is your transmission control unit (TCU) This is located under the passenger rug next to the main computer. I just purchased a reconditioned TCU for my 91 accord from an ebay seller called allcomputerresources. If he does not list a 91 accord TCU, then contact him to see if he can get one quickly. He will send it, and then you send the old one in as a core. My unit is running fine, and costs considerably less than from a dealer.