In the sentence "Śpię w moim łóżku" the noun "łóżko" is in the locative case (and "mój" just agrees with this case). Some further examples: "To jest moje łóżko" = "This is my bed" (nominative), "Nie lubię mojego łóżka" = "I don't like my bed" (genitive), "Pod moim łóżkiem jest potwór" = "There is a monster under my bed" (instrumental); see https://en.wiktionary.org/wiki/%C5%82%C3%B3%C5%BCko.
unfortunaltely hints are often confusing.
moim is locative masculine singular=locative neuter singular=instrumental neuter singular=instrumental masculine singular = dative plural masculine personal= dative plural not masculine personal
in this sentence moim is locative neuter singular