Inger comes to us from Denmark, and tells me that in Danish her name is phonemicized as /ˈeŋə/. The first vowel in Danish sounds rather like the English KIT vowel /ɪ/, which is handy as that’s the vowel English speakers will readily choose on the basis of the spelling:
(The final -r is unpronounced in Denmark, just as in England; see SCEPLog 5.)
Young Shin (영신) comes from Korea; the first syllable of her name, which can also be spelled Yeong, sounds in Korean rather like English young, /jʌŋ/:
(English shin /ʃɪn/ is a less close match for the second syllable in Korean.)
In English as in Danish and Korean, /ŋ/ is a fully-fledged phoneme which can appear without a following /k/ or /g/. This can be tricky for some non-natives to pronounce, especially when a vowel follows, as in lo/ŋ/ago or goi/ŋ/out.
There’s a general rule in English that /ŋ/ doesn’t appear before a vowel which is in the same morpheme (word-part). So finger, a single morpheme, is /ˈfɪŋgə/, while singer, made up of two morphemes, sing + er, is /ˈsɪŋə/. However, there are exceptions like hangar which is commonly pronounced /ˈhæŋə/ despite being a single morpheme. So a name like Inger might go either way. In my recording above, I said /ˈɪŋə/, approximating Inger’s own pronunciation.
Note that if we say Young Shin’s full name, Young Shin Kim, the /n/ of Shin may assimilate to the /k/ of Kim, giving Young Shi/ŋ/Kim: