I really appreciate your help. This form of language has haunted me for some time now. I like the second option, thus, I have written this way: "the thing which invigorates me, I am compelled to do." Would this suffice?
It doesn't sound very English to use this sentence order. It is better, in terms of the order, to say "I am compelled to do the thing which invigorates me". Similarly, the idiom is "You are what you eat", not "What you eat, you are".
I used "the thing which" as an example of the general rule that I stated - you shouldn't really use it in this sentence. When you say "the thing", it almost sounds like you are referring to something specific. The best way, in my opinion, of saying this is:
"I am compelled to do what invigorates me" or "I am compelled to do that which invigorates me".
But then... what do I know - I'm not a teacher!