Let's see, in terms of Daisy, a simple background check (if that's what you're saying Luther will do) will reveal almost nothing: driver's license, no tickets, no accidents, etc. No arrests. Very financially solvent family in Baltimore, etc.
Kiki has a pretty public life/record; lots of arrests for DUIs and drugs and, I don't know, one protest maybe, but no convictions. Maybe a stint or two in rehab.
Jack would be easy to find info on, and he's pretty clean too. West Coast or overseas for the last 10 years; dead cop mother, brother in an institution. If he's talented, Luther could probably find out that Jack was the secondary signer getting his brother committed and has paid for Nick's institutionalization for the last decade.
is that the kind of stuff you want to know? Let me know if you want more specifics.