First thing's first:

Я сыт по горло этой ерундой.

"Я болен и устал от этой ерунды" would just mean that you're ACTUALLY sick (like came down with a cold or something) and the second part would have pretty much the same meaning as "I'm tired of".

So, in hindsight, you can just say:

Я устал от этой ерунды. Though it won't be as expressive as "я сыт по горло".

There is one important thing to note though. "Сыт по горло" is quite strong an expression, in a lot of cases it could mean "I cannot take anymore of this..."