Transformers are quite a bit different than motors, my experience has been, not many people rewind them and if you do find a shop that will tackle it expect to pay dearly.
If you want to save the old girl.. Typing this as I think because it's been years....
If the transformer has metal covers remove them along with any bolts holding everything together. Take a good look at how it's put together and take notes on how the laminations are arranged and how/where the leads are connected to the windings. Throw the transformer in the burn pile, if you don't have a burn pile, fire up your charcoal grill using scrap wood and throw the transformer in, charcoal could get too hot and melt the copper wire. The idea is to burn off the coating they dip the things in so you can get the iron laminations separated. Work at this until you can pull the metal laminations out from the coil. Pay attention as to how they come out so you can put it back together.
Once you have the coil(s) free of the core, unwind the wire while counting the turns. You will have a coil of heavy wire which is the secondary then probably under that will be a smaller winding which is the primary. Keep one piece of wire from each winding so you can match the wire size at your local electronics supply house. Better yet go to a motor rewind shop for the wire and some of the tape they use inside motors.
Now the fire probably destroyed the original plastic form the windings were on so you will need to fabricate something to separate the wire from the iron core and provide a form/spool to wind the wire back on. If you can find some sheet pvc plastic, make a box that will just fit over the iron core. I am thinking some fairly thin plastic bent into a square box and a couple drops of PVC glue applied with a toothpick should make a pretty good spool. It would help if it had side boards to keep the wire from slipping off the ends. A couple more pieces of plastic and glue should take care of that. What you have just made should look like a square wire spool and be a snug fit over the transformers iron core. If you don't want to mess with spool ends/sideboards you can lay a piece of the tape under each layer crossways from the windings and loop it back over the end of the windings to hold the wire in place. The problem is this may make the spool to big to fit back into the transformer core. You will just have to play it by ear and do what you think will work.
Now it's just a matter of winding the wire back on the form and soldering stranded wire pigtails to the ends of the magnet wire. Use the tape you bought at the motor shop between the primary and secondary windings and also to secure the connections to the pigtails. The winding will need to go on in neat layers or you probably won't have enough room to get all the wire back on the transformer. If you can find a rotary counter that will count each rotation you can chuck the spool in your drill motor, rig up the turns counter and speed up the winding process.
You may be able to get it apart without the burning process but I never had much success without cooking off all the coatings. Like I said it's been years but this is the general idea and will work if you are willing to spend the time.
