The 10/22 does not have a dovetail rail. You just buy the proper mount for the rings you have. The 10/22 is drilled and tapped just like a big bore rifle. You can buy the whole set up for weaver rings for under $20.
Just return the rings you have and buy some weaver quad lock rings. They are under $10 if you have the mount. You can't make chicken soup from chicken $h!t.
People who don't have a 10/22 shouldn't give advice. it leaves you looking like a fool.
Ok, I think I made something work. I got quad lock .22 tip off rings that fit. It says up to 36mm lens it fits. The scope I have is a 3-9-40, however, they will fit around them. Considering its a .22 cal, recoil shouldn't be a issue and think everything will be good!!
I too have been looking at mounts, rings and scopes. Almost all tubes are 1". Then you have to match the mount to the rifle. Here are a couple inexpensive sets that look pretty good to me. I want a low set.
I tend to stick with weaver products for my scopes. They are like a 30/06 everyone has them. Rings and mounts never fail at home. They wait until you are 300 miles from home in the boonies somewhere to break. If your mounts and rings are weaver you can fix the gun and start shooting today.