Door bell transformers usually look like this: Often you'll find them stuck to the side of the circuit breaker panel thought they could be anywhere. It's possible that the transformer was removed (or perhaps was never there).

I just used the ring chime and have been happy with it. No external chimes. Requires the ring doorbell of course.

It should be doable with a little caution. I would do the following steps: 1) remove the doorbell and try to see where the wires go (probably straight in for a ways). 2) gently pull on the wires to see if you have any slack (this way you'll know whether you have to extend the wires or not to get to the face of the jamb. 3) This is the tricky part: drill a hole -- probably 1/2" diameter through the face of the jamb about where the intersection of your X is. You want this hole to just barely intersect the edge of the existing hole but not go right through the middle of it as that would sever the wires. You could have someone pull on the wires to bring them to the top part of the existing hole while you drilled to intersect the bottom of the existing hole. If you can do this successfully, then you can fish the existing wires to the front of the jamb with a wire hook if they are long enough. If they aren't long enough, you will have to extend them by soldering several inches of bell wire to each lead and then insulating the splices. Fish the new wires through the new hole and have them long enough that you can leave the splices in the existing hole as it might be impossible to get then to negotiate a right turn toward the front. An alternative to point 3 is to push the wires deep enough into the existing hole that you won't hit them when you drill through, and then drill through the center of the existing hole (it will make the re-routing of the wires easier). Once the cross-bore hole is drilled, use a wire hook to gently coax the wires back out of the hole and then fish them to the front. If you mess this up and sever the wires, then your only option will be to temporarily remove the piece of jamb trim that the X is on and do what you have to get at the wiring; then repair the wires, and extend them several inches. That's the easy part; reattaching the trim, filling the holes, and repainting and re-caulking are the not-as-easy part. Do those ideas make sense?

pookieman- You're not going to find a doorbell that works without some power source. 2. Enlarging the hole enough to get at the wires will only work if it's big enough to splice extensions on to the existing wires, and that would require stripping insulation off the ends of the wires in the hole, soldering or crimping new extensions on, and insulating the splices if necessary... which would likely require a bigger hole than you want, and some fairly good DIY skills. 3. It might be easier to cut into sheetrock on the inside to get access to the wire ends and repair them, and then do a sheetrock repair. 4. The only other option I can think of is to fish a new wire into the wall and run it back to the transformer.