I looked at you pictures and it's similar to what I have built, only I powered mine from 14.5vdc from a car.
With your 15 vac transformer, use a full wave bridge, then connect 2 or 3 strings of 4 LEDs with your LM350 in the current configuration, your transformer doesn't appear to be able to power 5 in series, so add or subtract 2 LEDs.
I assume you got your LEDs off the internet and they are very similar to these: Color temperature:6000-6200K, Forward Voltage: 3.0V~3.2V, Forward Current: 350mA.
The current through a LED is what counts, too much and they die, too little and they dim or go out. On these, hold the current at 350ma and they are happy and bright as long as you get rid of the heat.
OH YEA, The 3.57 ohm resistor need to be 1/2 watt, recommend 1 watt