If you want to replace the drive yourself, it will just be the cost of the drive. You can look at drives at OWC and at Newegg. You want a parallel ATA (PATA) drive and NOT a Serial ATA (SATA) drive. You can get a drive up to 320 GB for an iBook G4.
For instructions, You can look at ifixit:
http://www.ifixit.com/
If you don't feel up to doing this, I have heard DT&T highly recommended. I believe they have free estimates and may also have a price list for some things:
http://www.dttservice.com/index.html
However, there is no reason why you can't use an external FireWire Drive as your boot drive and just ignore the internal one, which seems to be failing or has already failed. This might actually be a good course of action for the present. You can continue to use the iBook and be sure that everything else is working, and then can contemplate installing an internal drive at a later time.
If you don't already have an external drive, I would recommend buying a PATA drive and putting it in one of OWC's Mercury on the Go FireWire enclosures, which you can often find at a discount on their garage sale page. The drive will work from the FireWire cable, so the iBook will still be portable. Should you at some later time decide on replacing the internal drive, you can just take the drive out of the enclosure and install it. It will already have all your stuff on it.
You can probably install 10.4.11 yourself by selecting the external drive as the destination, but if you have access to a genius bar, no reason not to let them show you how so you'll know in the future.
Good luck!