Learn Swedish: Mönstret var monotont, men vi kämpade på ändå. - The pattern was monotonous, but we kept on struggling anyway.

The particle på adds a continuative meaning to the verb. While kämpade alone means "struggled" or "fought", kämpade på means "kept on struggling" or "soldiered on". It emphasizes that the action continued over time despite the monotony.

Why does monotont end with a -t in this sentence?

It has to agree with the noun it describes, mönstret (the pattern). Since the root word is ett mönster (a neuter noun), the adjective monoton must take the neuter suffix -t.

Why doesn't the verb come immediately after men? Doesn't Swedish use a strict verb-second (V2) rule?

It does use the V2 rule! However, men (but) is a coordinating conjunction. These conjunctions act like a bridge between two separate main clauses and don't count as the first position. So, the second clause starts fresh with the subject vi in position 1, followed by the verb kämpade in position 2.

Could I place ändå earlier in the second clause?

You could, but you have to respect the V2 rule. You cannot say men vi ändå kämpade på because the verb must be in the second position. You could say men vi kämpade ändå på, placing the adverb after the verb but before the particle. However, leaving it at the very end of the sentence is very common and sounds perfectly natural here.