Why doesn't "wish" have the letter "n" in it?

Solution 1:

Not my field, but I'm going to guess that this is an instance of the Ingvaeonic nasal spirant law, under which an /n/ before a fricative is dropped in Old English, Old Frisian and Old Saxon.

The article linked names several other English words which exhibit this change, including

• mouth, compare Ger Mund
• us, compare Ger uns
• tooth, compare Ger Zahn
• other, compare Ger ander
• goose, compare Ger Gans (and note that gander, where no fricative is in play, retains the /n/)
• five, compare Ger fűnf
• soft, compare Ger sanft

The article also explains why some words, such as month, tenth, and answer appear to evade this law but in fact do not.