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I'm much shorter than you with lower back problems, and have an expensive chair at home so can't recommend that to you. But in situations where I can't use that chair, I find that two things help immensely. One is proper lumbar support, the other is a footrest.
A lot of lumbar support pillows are either weirdly bulky and rigid, or else go flat immediately. The best one I've found (either in the US or UK) has been this one. It's not small by any means, but it works well for me even with a dining chair. The elastic is useless after a few months but not necessary anyway.
The other thing that helps a lot is a footrest. Proper alignment means you want you want to be able to sit back in the chair, with your thighs roughly parallel to the floor and your feet flat on the ground. If you're short, that doesn't happen. With a small footrest you can adjust the chair height so that it's appropriate for the desk, but then still have support for your feet/legs.
I hope someone recommends a good chair, I just thought I'd pass along these tips as they've helped me cut down on a lot of back pain.