Well, thieves, cads and con artists generally avoid notoriety, it's professionally detrimental most of the time. Whaykim, however, was known about Korvosa before his disappearance so I'd like to capitaliize on his reputation rather than her own for recognition. It would be well known that she is his operative about town.
I left the circumstances of Whaykim's disappearance and his current whereabouts completely undefined so that you would have the opportunity to capitalize on these vagaries to create something suitable as you see fit; a mystery of sorts for Scritch to investigate and a method for you to motivate or direct my character as time goes on. Perhaps his disappearance is related to the undead in some way. There may also be other parties interested in his return or those interested in ensuring his return does not occur.
Even if she did not discover a specific clue each day or spend time specifically investigating his disappearance, if Scritch is working with, gaining exposure to and acquiring skills from investigators, she is making an effort to find him. She might even stumble across something immediately relevant to Whaykim during the course of working with the group but I don't even see that as necessary. I suggest that it is more about her intent than the fruitfulness of her activities. If she decided to abandon her efforts and head to Magnimar to open a massage parlour, that would be a different story.
I'm not entirely sure I understand your objections to the use of Whaykim's library or how that relates to her motivations or lack thereof. And as far as a particular useful skill, it's hard to tell from just the fluff but skills, all of them pretty much, are what she brings to the table. She's no slouch in combat and, being melee focused, will fill a hole you mentioned the party having. Here she is, almost done except for equipment.