I think that Emel gözlüğünü kaybetti could mean any of "Emel lost his (someone else's) / her (someone else's) / her (own) glasses.
And Emel'in gözlüğünü kaybetti could mean any of "He (someone else) / She (someone else) / She (Emel) lost Emel's glasses", but probably only one of the first two, much as "She lost Emel's glasses" in English would likely not be used if she = Emel.
I think that if you want to stress that she lost her own glasses, you could say Emel kendi gözlüğünü kaybetti "Emel lost her own glasses".
Kaybolmak = to be lost. ( recently lost ).
Kaybetmek = to loose (lost long back in time or Its been some time you lost something)
Mesela (kaybetmek) : I lost my phone = Cep telefonum kaybettim
Mesela (kaybolmak): My cell phone is lost = Cep telefonum kayboldu ( Notice is lost in the sentence which indicates its lost right now or recently).
His glasses - onun gözlüğü Why the extra 'nü'
Accusative case, since "glasses" is both definite (they are not just any old glasses; they're Emel's glasses) and the direct object of the verb "lose".
The -n- is a buffer consonant since you can't have two vowels next to each other in (pure) Turkish words.