This H is called Damper rod stopping tool, and size between 11 and 12mm is just right. I have 12.3 - 13.3 tool and is slightly to big. Damper rod nut size is 14mm.
1l of oil is just enough.
Tricky part of fitting back spring collar is to hold damper rod in place , as it's dropping down below top of the spring.
Are you using stationary spring compressor?