Learn Finnish: Kunnostitko sen senkin, jollaisesta me haaveilimme? - Did you restore that sideboard, the kind we dreamed about?

The case is dictated by the verb haaveilla (to dream). In Finnish, when you dream about something, the object of the dream must take the elative case (-sta/-stä). Since the speaker dreamed about "the kind of" sideboard, jollainen takes the elative ending and becomes jollaisesta.

Is the word senkin just the pronoun sen (that) with the -kin (also/too) clitic attached?

It looks exactly like it, but no! In this sentence, it is the accusative form of the noun senkki, which means a sideboard or dresser. The sentence is talking about "that sideboard" (se senkki). Because it is the total object of the verb, both the pronoun and the noun take the accusative -n ending, giving us the repetitive-looking phrase sen senkin.

What is the difference in meaning if I used the basic relative pronoun josta instead of jollaisesta?

Using josta (the elative of joka) would mean "the sideboard which we dreamed about" — implying the person restored the exact, specific physical piece of furniture from their dreams. Using jollaisesta (from jollainen) means they restored a sideboard of the same kind or style as the one they dreamed about.