Translation:My child loves English songs.
Am I correct in thinking that 英语 refers only to the language, or can it be used for nationality as well? I got this wrong the first time because I wrote 英国的 - as in songs from England (which would also generally be called "English songs" in English). I learned 语 is language, 国 is land or country, and 文 is culture (and language), so English (language) is 英语 or 英文, and English (person or something from the country) is 英国～.
"~语" only refers to languages :) However if you really use "French song" to mean "the song from France," it can really be "法国(的)歌," though this sense is less common. For "~文," it generally just means languages. "法语" and "法文" are interchangeable for most speakers. The others you mentioned are all correct :)
I think "songs in English" should be accepted. Honestly "English songs" isn't a good translation anyway. If you say "English songs" it sounds more like songs from England specifically, 英国哥. American songs and Kiwi songs can also be 英语歌 but nobody would call them "English songs" even if that's technically correct.