A failing ABS accumulator shouldn't affect the bleeding of the brakes, just the operation of the brakes
as it provides the power assist. But, if before your brake job you had sonstant firm brakes, you should have some time before you need to panic.
For the front brakes, presuming you know how to bleed the calipers such that no air reenters the caliper, then its just the usual:
setup: loosen bleed valve ever so slightly, reclose finger tight, hook up clear tubing (to see the air bubbles) to the bleed valve and the other end submerged into a tin can with about 3/4" of brake fluid.
1. have assistant (kids are good for this) hold down pedal,
2. open valve enough for fluid to flow out (not too much)
3. bleed for a few seconds
4. close valve
5. release pedal
6. repeat steps 1-5 until you have bled around 1/2 cup and no more air is seen going out of the valve into the tubing. Be sure to refill the reservoir occasionally.
Now the rear brakes are different; with those, you have to turn the key on to the run position (to keep the pump running to charge the accumulator).. and you don't need anyone to press the pedal, although I suppose they could to help.