This is the 9382nd most frequent Polish word.
uśmiechu
"Uśmiechu" means "of a smile" (genitive singular form of "uśmiech").
The genitive case is used here to describe the 'shine' being 'of a smile'.
Widziałem blask uśmiechu na jej twarzy, która promieniała radością.
I saw the shine of a smile on her face, which radiated joy.
The dative case is employed to denote 'adding' something (a smile) to the speaker.
Twoja pochwała dodała mi uśmiechu, za co jestem bardzo wdzięczny.
Your compliment added a smile to me, for which I am very grateful.
The instrumental case is used here with 'with which you gifted me', referring to the smile.