If you can get hold of one get a timex 2020 i use it and it's great never lets me down. (but i think you'll be hard pushed to find one).
you can always used your PC to load porgs into ZX81 there are a handful of programs that will either play tape files directly (i use wintzx) or convert them to Wav's for you to play back using software of your own choice. look on WOS for links to these programs.
If you don't mind asking, where do you live? If you're in the U.S. like me, I might have some of the things you need. The TS1000, for example, was one of them. Regarding the cassette player, here in the U.S. you can still go to a Radio Shack store and buy the ones they used to sell for 8-bit computers, new in the box. I use one of those and a Timex Sinclair 2020 recorder and both work great.
For more ideas on how to use the ZX81, check Simon's website at zx81stuff.org. The idea of using the PC's soundcard is a good one too.
However the stereo ones are noticably worse.
I found it can help to load progs when You have BALANCE. Then I simply put whole signal into only one channel - then it works better.
When my ZX81 come by post, I will test regular PC as a tape (hehe).
It works OK with Spectrum (however with amplifier) so I suspect it will be OK for ZX81 too.
Basses/trebles corrector would be helpful too.
Would it be about 8 minutes? I found such an info somewhere.
So my rough reckoning if you where to save all 0 bits it would tale 5 1/2 minutes for 16K where as all 1 bits would be just shy of 9 minutes. So i guess the real answer is between 5 1/2 and 9 minutes depending on the bit pattern within the program.
I would be happy to build some .p -> fast load wav converters given the relevant information about the required waveforms.
I have to ask: Why not use an emulator? I personally love using the real old hardware - but sometimes I think that life's too short