It would be posible to say, "I like she who is honest", but a) it sound archaic and b) it really implies that the "she" means "any female".
I'm not entirely sure of the exact meaning of your phrase, but "I like her; she is honest" or putting in a conjunction like "I like her as/because she is honest" sounds better to my ears.
Good job we agreed, really! There should be a term for this phemonaminum. It needs to be a combination of words like "Simultaneous typing". How about... Typletaneous simping?
Hmm or Simulaneously typed submission - Simultypission. Erm.
An acronym, mebbes? Posting Including Simultaneous Submission. Hmm.
"I like her who is honest" would be grammatically correct (even though it sounds awkward) because "her" (accusative) is the object of "like", and "who" (nominative) is the subject of "is".