A couple of things I learned when chair shopping:
i) The phrase "high back" means nothing.
I found that the back height varied as much as 40cm between chairs described as "high back".
ii) Stated height measurements are not necessarily comparable because they don't necessarily measure like for like, nor do they necessarily mean what you'd expect.
Back height is the distance from the top of the seat to the top of the back, right? After all, that's the back height measurement that's actually relevant.
Maybe it is that distance, maybe not. Maybe it's the length of the entire back section, which might start below the bottom of the seat. That's the case with the Ikea Markus and the difference is 8cm (i.e. the distance from the top of the seat to the top of the back is 8cm less than the stated back height). Which is still fine, because from seat top to back top it has the highest back I've found on any chair.
The only way to be certain of a chair's size is to measure it yourself. Ideally, sit in it yourself. A chair that's the right size might be uncomfortable or shoddily made. Even reviews and recommendations might not fit you, because it's a very subjective choice. For example, I recently bought a chair that was horribly uncomfortable for me. A wooden stool would have been better. MY old broken chair with just the seat was better. I sold the new chair to an acquaintance for £20. She was delighted with it and finds it very comfortable. Then I did what I should have done in the first place and bought a Markus.
For what it's worth, I'll add another recommendation for Markus. It's a little over your £100 budget, but I've sat in £250 chairs that were worse. I tried over a dozen chairs in the £100-£250 range and the Markus was the best.
