I would want two players in each position. So, of the players I would keep (And restricted to naming each player in the position as I see their strongest.)
Goalkeeper: Davis, New Signing
Right Back: Richardson, New Signing
Left Back: Fox, New Signing
Centre Back 1: Hooiveld, Martin
Centre Back 2: Fonte, New Signing
Right Midfield: Puncheon, New Signing
Left Midfield: Lallana, New Signing
Central Midfield 1: Schneiderlin, Cork
Central Midfield 2: Chaplow, New Signing
Striker 1: Lambert, Do Prado
Striker 2: Lee, Sharp
So. That would stipulate the following new signings:
Goalkeeper: Alex Smithees
LB: ??
RB: Nathaniel Clyne
CB: Ben Turner
RM: Robert Snodgrass
LM: Matt Phillips
CM: Peter Whittingham
I would also make a move for Jordan Rhodes. Ensure we have ample options up front. With the larger bench in the top flight we could be far more flexible. Also sign Gael Birgirimana from Coventry, he is going to be a superb dominant CM in years to come.
Notable redundancies:
Harding
Bialkowski
Butterfield
Hammond
Connolly
Seaborne
Jaidi
Barnard