I'm sure this has been covered before but I can't find it. I tried fitting LED lights to my trailer but the indicators flashed weakly all the time.

I have also had a hired car trailer with LEDs and the car couldn't detect it and didn't disable the reversing sensors.

I've considered a resistor in the earth cable, will that work? Or some other solution?

I think you have to fit a ballast resistor into the circuit if you have LED lights on the trailer. I have Led lights on my trailer and the car does not recognise that the trailer is attached thus not disabling the parking sensor. The ballast resistor is on my "to do" list

If its anything like the RR I used to have, then the light control module would pulse all the lights in the car intermittently with a small current to check whether they were present/working or not. For normal filament bulbs, the small current would never be enough to light the bulb. With LED's even a small current over 40mA is generally enough to get the LED to begin to light up dimly. ( In simplistic terms, LED's brightness is more current dependant than voltage dependent )

If the car was switched off for more than 20 mins, then it would be a short blink every 5 seconds. If the car was awake so to speak, then it would pulse on off on off very dimly, changing every second or so.

You may need to get canbus LED bulbs. These have a small simple circuit added to them that fools the light module into thinking the LED's bulbs are like filament ones.

That being said, I never tried a set of can bus ones on my trailer to see if the pulsing went away.

I found I had the same problem (faintly flashing trailer lights) about 2 years ago when I bought a trailer with LED lights. On investigation I found that its a common problem in Australia with European softroaders (Porshe, VW, BMW, LR, etc). There is a company called Linear Electronic Design (see link) which advertises a box that prevents the flashing but its expensive. Also I tried Landrover who quoted a similar price for a device they said that they had. I decided to just change the LED's for a set of filament lights (about $60) form local auto store. Worked fine.
http://www.linearelectronicdesign.com/buy.html