Norm Dresner says...

During a move I broke one of the wooden legs of a (circa 1940) wooden desk
right where the round turned leg was attached to the square top that forms
the corners of the case. Of course it's not a clean break, though it's not
that jagged either, and I can fit the two halves together by hand quite
closely.

Dowel joints have to be exceptionally accurate even under normal
circumstances, but in this case it would have to be perfect. Even a few
thousandths of off would be too much considering the jagged edges have
to meet perfectly. I strongly suggest you forget about that scheme. It
won't work with the tools and techniques you described. The obvious
course of action would be to just glue and clamp the break without any
reinforcement. That's what I would do. It might even be more or less
as strong as the unbroken wood that way, but in any case it would only
break again if you had another major accident. I wouldn't use epoxy
either. The bad thing about epoxy is that once it hardens, it is there
forever. You will have a real mess on your hands of you have a false
start with epoxy. I would use hide glue because it is reversible. If
it doesn't go together right, you could (with some difficulty) take the
joint apart again and reset it. If you feel you must reinforce the
joint, do it after the break is glued together and square. You could
put a spline in the back side as someone else said. If the top is not
in the way or can be removed, you could drill from the top with an
appropriately long bit and add a dowel that way.