Here's what I've got so far. I'm stuck on how to proceed. I believe I need to plug back into Euler's formula, but I'm not getting what I'm looking for by doing that. Where is the denominator of $3$ coming from in the result? Can you please check over my proof for any incorrect statements, and help me move forward?
Proof. Let $G$ be a connected planar graph with at least two edges, and a cycle with length $\ge 8$. Pick a crossing-free embedding of $G$; this embedding has $f$ faces. By Euler's formula, $$f = 2 - |V| + |E|\;.$$ We calculate the sum of the degrees of the faces in this embedding. On the one hand, the sum of the face degrees is $2|E|$ by proposition. On the other hand, since there is a cycle of at least length $8$, the sum of the face degrees is at least $8f$.
That's where I'm stuck at. Any ideas?