Whilst in most cases, I'm clear using who or whom but this is one of the sentences bothering me!
Whom/who did she invite?
Whom did she invite looks very very natural to me. The general rule (this will work in most of the cases, no exceptions please!) I remember is... be yourself in that situation, and if 'I' fits, it'll be 'who' and if 'me' fits, it'll be 'whom'.
Some examples where this rule works as a breakthrough...
Whom this matter is concerned to? - The matter is concerned to me and not I so whom!
Our madam wondered who broke the glass - I broke the glass and not me so who!
That way, She invited me and not I so it should be - whom did she invite, shouldn't it? Also, if you take it other way - She invited him and not he.