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If I read this correct, you intend to make a new driveway from the front of house to the gate leaving the stairs at the side of the house in place as they are, so you won't have any more run off than you have now. ??
If you are concerned with water, is it possible to raise the drive so water runs to the street.? although that in turn may involve changing the stairs, and, I must warn, you some councils require a grated drain at the fence line to catch any runoff and piped direct to kerb to stop runoff going across the footpath.
To clarify I would be removing the stairs and sloping the land down towards the gate where this photo is taken.