Learn Icelandic: Þú þurrkar hana upp með tuskunni.

This is a key rule for phrasal verbs in Icelandic. When the object is a pronoun like hana, it must go directly after the verb and before the particle (upp). If the object were a full noun, the particle would usually come first, like Þú þurrkar upp mjólkina (You wipe up the milk).

The translation says "it", but doesn't hana mean "her"?

Yes, hana is the feminine accusative form of "she" or "her". In Icelandic, you must use the pronoun that matches the grammatical gender of the noun you are replacing. Here, whatever is being wiped up must be a feminine noun, so we use hana to mean "it". If it were a neuter noun, we would use það.

Why is tuskunni in the dative case here?

The preposition með takes the dative case when it means using something as a tool or instrument. Since you are using the rag to do the wiping, the noun changes to its dative form. If með meant carrying or bringing something, it would take the accusative case instead.

How do we build the word tuskunni from the base word tuska (a rag)?

First, tuska changes to its dative form tusku because of the preposition með. Then, we add the definite article suffix for dative feminine nouns, which is -nni. This gives us tuskunni (the rag).