Give the ring to Frankie after rescuing her. She will respond that she has to think about it. Stop in to see her before going to the police with the diamonds and show them to her. You'll receive a kiss and she will accept your proposal afterwards.
Hints: Paying the landlord
Give the betting slip to the barber. He'll ask you if you want to make a bet. The odds are 25:1. You will now have enough money to pay the landlord.