Answer:
Kitty, a pretty girl called after a cat, couldn't make it to school that week; nevertheless, she couldn't fail to check up her friend Senami. Senami was at home, having fun with her cat Kitty, which she called after her friend.
The appositive phrase is "a pretty girl called after a cat". It shows a phrase segregated by commas, which illustrate the noun or supplies extra information.
The conjunctive adverb here is "nevertheless". It seen to link two independent clauses as one sentence, communicate the connotation of contrast (Kitty couldn't go to school, but she checked up on her friend).
The relative pronoun in this case is "which". It is relative since it speaks of "her cat Kitty", and it brings in a dependent clause.
The compound sentence employed in the short paragraph is "Kitty, a pretty girl called after a cat, couldn't go to school that week; nevertheless, she couldn't fail to check up on her friend Senami." It uses two independent clauses: "Kitty, a pretty girl called after a cat, couldn't go to school that week" and "nevertheless, she couldn't fail to check up on her friend Senami".