"How do you slog through something you truly hate?" - I don't.

When signals that a role is not aligned with my needs start cropping up, I begin searching for a new role passively, and as the situation develops I speed up my search.

"I felt like a complete failure because I didn't put in 100% and I felt like I let my mentor down" - to thine own self be true. I have failed to put in 100% at some jobs, and sometimes i regret it more than others. I have narratives that legitimize my laziness or lack of commitment based on some previous slight from the company, or a missed promise on their part, but I hold myself accountable.

"How do you overcome something like that and push on? It does not always seem like a sure thing" Resilience is a wildly varying trait of folks, and depends on your emotional and mental state. "First world problems" are a great example, one when is socialized at a certain comfort level, missing that causes distress. Some working conditions are truly untenable, in which case do what you have to do, but otherwise do the best with the situation you're given.