Need a 24V AC power supply, 850mA

That page is not responding, but...it's AC, so polarity is not important. If the connector is the same and the current capability is at least as high as the unit you have, it should work.

Another option, if you are up to it, is to get another connector and replace the one with the broken pin, or, with the correct strain relief, hard-wire the cable to the supply. Only do this if you are somewhat knowledgeable about electronics, or maybe get a friend to do it, or some hobbyist teenager who knows electronics.

RS Componnemts

..have one (search for Part number 494-8776) but you need to find someone with an account. This particular item is sold as a power supply for CCTV cameras so perhaps your local Alarm system retailer will have one?

The nearest I could see at Farnell was only rated for 750mA (not 850mA) which isn't enough.

Many thanks for the replies.

I've uploaded a picture of my (now broken) adapter here:

If I purchase the power supply mentioned at RS Components, could I just cut the plug from my lights' cable and attach that cable to the power supply?

The short answer is yes. I haven't looked at the full data sheet but the photo shows that the power supply has an output cable probably with plug on the end of some sort like most mains adaptors have. You would either need to cut that plug off and join up the wires or find a matching socket to fit on the lights. If you cut it off you could use a regular terminal block to join the wires or use a soldring iron and some heat shrink sleeving. Since it's AC it won't matter which wire connects to which. I wouldn't try and open the power supply to try and connect inside. It's probably sealed anyway.