There is an inherent issue with ZX81s and LCD tvs. If its an issue 1 Zeddy, then its likely to have a ULA that provides no 'back porch' signal, which, translated to English usually means that the display becomes very dark.
Later versions of the '81 (v3 onwards) had a better ULA which gives a better picture on modern TVs.
The problem is that someone could quite innocently sell you an '81 as tested with pictures showing it on a CRT - but when you get it, it looks very poor on your LCD - its always going to be a gamble.
There is another issue too... When the ULA starts to get warm, the signal starts to drift, so you have to keep retuning the tv - old CRTs seem to have some 'aft' setting that logged onto this drift and auto-corrected it. LCDs (or all the ones I've used) sdo no such thing.
You could go the ZX Vid route - available from RWAP - that replaces the modulator, adds the required backporch signal and gives you composite, or you could buy a ULA2 (but then you haven't got a true ZX81).
Now, with all this in mind, I can sell you a ZX81, but wont be able to guarantee it will work on *your* tv, but could show you piccys of it on mine. If you are interested, drop me a PM and we can discuss a price or a possible exchange or r3tr0 items