The correct version would, I think, be "None of us knows what he's doing", although both the feminists and the languagelog folks would complain about that.
gcide defines “none” as “No one; not one; not anything; – frequently used also partitively, or as a plural, not any.” and also offers “None of their productions are extant. –Blair”, both purportedly from 1913 Webster. So it seems that the pronoun can be interpreted in either number as needed.
"None" == "Not one", and therefore singular.
“USAGE It is sometimes held that none can take only a singular verb, never a plural verb: : none of them is coming tonight, rather than : none of them are coming tonight. There is little justification, historical or grammatical, for this view. None is descended from Old English nān, meaning ‘not one,’ and has been used for around a thousand years with both a singular and a plural verb, depending on the context and the emphasis needed.”
You’re right though that it’s weird to mix both singular and plural verbs for the same subject.