Andrew, I'll echo other above comments on safety (lack of rigid mounting of board, heatshrink, fusing, appropriately rated wire on the primary side, etc).
>Off-topic ON<
Is there any reason for so much "dispensable" [(25*1.414 - 2*0.7 - min Vin-out) * I] power dissipation when only aiming at 9V output..??
(transformer secondary could be rated down to 10VAC without issue..)
>off-topic OFF<
While I agree that this design has excessive dissipation in the regulator, personally I'd be a bit more conservative than 10VAC, that leaves zero margin.
I'd aim for:
Vout 9VDC (let's say 9.6VDC for "fresh battery")
+3V LM317 dropout voltage (12.6VDC)
+2V ripple voltage (14.6Vpk)
+2x1V Vf - 0.7V is marginal for silicon rectifiers (16.6Vpk)
/Square-root 2 (~11.74VRMS)
/0.9 to allow for 10% mains voltage variation - don't want it dropping out in summer/winter when everyone turns on airconditioning/heating (13VRMS)
A 13V secondary might be a little hard to find, rounding up to the next common value of 15VAC allows for smaller bulk filter cap, or better, an RC pi filter before the regulator (I'd split the dropping resistance between positive and negative rails for better common-mode rejection - make this closer to "ultra-clean" - something I don't think this design has any claim to).