Depends on what you want to do. I ride reining horses I'm 5'9" around 126lbs and my currently horse is 14.1hh but reiners are small we keep them close to the ground so they can get into it when we ask for a stop they stay balanced during spins and circles. So look at your discipline, the breed (depend on the size of the bone) and how big of a horse you want to own!
This can be tricky because depending on you and the horse it changes. I look more proportional (I am 5'7") on my 17.1hh horse than I do on my 16.3hh horse
It depends on the build of your horse. I'm 5'10" and have two horses that are 16.2. I look a bit tall on my TB and my QH fills my leg up much better because of how deep she is in her heart girth.
I am 5'8 too. I have a 16 hand horse that I just am a little taller than but I look good on him. Just depends on what you are wanting to do.
I'm your height & my horse is 16.3. I would recommend that & up
A horse that takes up your leg is what's important. Nothing worse than having to hike up your leg to squeeze them. My 17hh guy doesn't take up my leg because of how lanky he is and I'm 5'3.
I'd say around 15.2+, but it also depends on your build.