Thank you for the replies so far.
Here's more details:
The printer uses a 24VDC PSU, output current unknown currently.
The LEDs are white ones in the form of a roll that I bought from eBay. They were called "Cob strip LED tape 24V". They incorporate LEDs and resistors and are rated at 24VDC and 0.5A per metre of strip. I've used around 32cm of strip total. They are self-adhesive with marked cutting points, and so I cut off two ~16cm lengths and adhered them to uprights on the printer. Each length is connected by two wires directly to the printer's PSU output.
I have some LM317s on the way for something else that I can try, I imagine I will lose a couple of volts on the output from the 24V input, but the LEDs should still be reasonably bright?
A couple of years ago I bought some SM2082 ICs from China for a project that I never got around to. Would they be suitable or is the NUD4001 preferable?
I hadn't considered overload. I need to take some current readings, but for now I can say that while the nozzle and bed are being kept at their set temperatures, the 18Hz signal is still present even with the white LEDs disconnected. I think the PSUs have overhead as some people add higher wattage nozzle heating elements to this model without problems.
I've attached two photos. One shows the white LED strip in the upright post, and the other shows the mainboard.
On the mainboard, one of the red LEDs lights up when the nozzle is being heated, and one lights up when the bed is being heated. They are solid red until the respective parts hit the pre-set temperatures, at which point they flash rapidly as current is switched on and off to keep the temperature at the set point.
The flashing rate of those red LEDs appears to match the white LEDs flicker rate, and the white LEDs flicker when the red ones on the mainboard begin to.
