This is a tricky sentence to parse, because that first for (the one after fight) actually serves two different purposes.
On the one hand it acts as a preposition in the idiomatic expression fight for X, meaning struggle to achieve X, where fight may be either a verb or a noun.
Superman fights [preposition phrasefor Truth, Justice and the American Way]!
Superman’s fight [preposition phrasefor Truth, Justice and the American Way]
In the sentence at hand, however, the object of that preposition is not a simple Noun Phrase but a subordinate clause of the form for A to do B, where for acts as a complementizer—in traditional grammar, a subordinating conjunction:
What the administration wants is [complement clausefor the states to sign on to the Medicaid expansion].
which might be paraphrased as
What the administration wants is [complement clausethat the states should sign on to the Medicaid expansion].
When that nominal complement clause headed by the complementizer for becomes the object of the preposition for, the two fors are collapsed into one.
... a fight for X
replace X with for the states to sign on to the Medicaid expansion
... a fight for for the states to sign on to the Medicaid expansion
collapse the fors
... a fight for the states to sign on to the Medicaid expansion
This is not ungrammatical; it’s just one of those shortcuts nobody notices.
The sentence may be paraphrased:
The Administration needs to engage ... in the effort to make the states sign on to the Medicaid expansion, among other things.