Learn Icelandic: Mamma er inni á baðherberginu.

The preposition á triggers the dative case when describing a stationary location. The base word is the neuter noun baðherbergi. To make it definite (the bathroom) in the dative case, we add the neuter ending -nu, resulting in baðherberginu.

Doesn't á normally mean on? Why is she on the bathroom?

Prepositions rarely translate word-for-word. In Icelandic, the preposition á is specifically used for certain rooms, most notably the bathroom. So while English uses in, Icelandic uses á baðherberginu.

What does inni add to the sentence if á already tells us where she is?

The word inni is an adverb meaning inside or indoors. Saying inni á baðherberginu emphasizes that she is specifically inside the enclosed room, though just saying Mamma er á baðherberginu is also completely correct.

Since the base word is baðherbergi, does it lose its final letter before adding the definite ending?

No, it keeps it! Neuter nouns that end in -i, like baðherbergi, keep that vowel in the dative singular. You simply attach the dative definite ending -nu directly to the end of the base word.