Switch to stay ON until power removed (24V)
How about this?:







simulate this circuit &ndash; Schematic created using CircuitLab

Your post mentions transformer but it must be more than that to power the LEDs. I'm assuming it's a PWM supply of some sort. If that's the case each pulse will be full voltage and we can 'steal' some of that to charge a capacitor to make our power supply. C1 is charged up by the PWM pulses via D3 (to prevent discharge) and R1 (to limit current draw). You'll need to calculate the values of these components to suit your relay coil. 

Pressing S1 will energise the relay and latch itself on while turning on the cool LEDs. When the PWM source is switched off C1 will discharge through the relay coil and reset the circuit.

If the external switch is a regular dimmer switch then this circuit is unlikely to work as the LED power supply is, more than likely, a simple transformer and rectifier.