Learn Finnish: Lukekaa tämä suosituskirje, koska suosittelen häntä lämpimästi. - Please read this letter of recommendation, because I warmly recommend her.

Because the verb lukekaa (read!) is in the imperative mood. When you give a direct command, a total object drops the normal -n accusative ending and looks exactly like the nominative case. If this were a normal statement like "You read this letter", it would be Te luette tämän suosituskirjeen.

Why is häntä in the partitive case here?

The verb suositella (to recommend) is a partitive verb. The person, thing, or action you are recommending must always take the partitive case, which is why hän becomes häntä.

How do we get the adverb lämpimästi (warmly) from the adjective lämmin (warm)?

To form an adverb, you add -sti to the inflectional vowel stem of the adjective. The basic adjective is lämmin, but its stem undergoes a consonant gradation-like change to lämpimä- (which you also see in the genitive lämpimän). Attaching the suffix gives us lämpimästi.

How could I use this lesson's mitä + superlative structure to call this "a most warm letter of recommendation"?