The transformer you have should be OK. Just use the rectifier bridge to drive the LEDs only. Put a filter capacitor on the output of the rectifier bridge (watch for polarity). A few hundred uF should be enough since we're only dealing with a current of about 20 mA. You only need a one-amp rectifier bridge (those are about the size of an 8-pin DIP package). In fact, even an IN4001 diode should be OK if you have a filter cap of around 470 uF or more (make sure to use one rated for 25VDC or better). You should have about 15VDC across the capacitor so a 470 ohm resistor will give you a current of roughly 18 mA if you connect the two LEDs in series. The low-voltage alcove light remains connected straight to the AC output of the transformer.