"I do not like sports very much."
I found this article which goes some way to explaining the meaning of では. http://selftaughtjapanese.com/2015/02/26/japanese-particle-combination-%E3%81%A7%E3%81%AF-de-wa-and-%E3%81%98%E3%82%83-ja/
I don't think there is such word as 好きない. For な-adjectives, the present negative form is "ではない" or "じゃない". Usually です is used at the end if you don't want to sound too casual. If you want to be really formal and polite, then replace ~ない with ~ありません. No matter what, では or じゃ has to be there for な-adjectives, with では more formal than じゃ.