Since 5.2 came out, I've had a few instances where i use my zhonyas, but it doesn't go off. Normally, this leads to death, and it seems like it doesn't matter if i click it once or 5 times, it doesn't go off. A few of my friends have seen this too, and i don't know how to recreate it, it's seemingly random. If anyone else has had this error, it'd be nice to know.
It seems as though you could be having a similar problem to the clicking and ability issues being delayed. It seems like anything that can be "activated" is having a delay. I've noticed this with my skill shots as well as my BotRK.
I would like to say I have the same problem, but I was having this problem where I would keep pressing Zhonya's while getting gangbanged and I die before it goes off, so I am not much to have on your side.
