The fare quoted is a Super Off-Peak Return so flexible and available to buy at any time.
eastcoast.co.uk gives me £22.05 on the 1103 from the cross arriving 1330, and the return at 2005 arriving 2227 for £16.65. These are Advance fares.
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Alternatively for flexible tickets, split at Peterborough. A London Terminals - Peterborough Route FCC Only Super Off-Peak Day Return is £18.00 (10% off from FCC's website), and a Peterborough - Gainsborough Route Any Permitted Off-Peak Day Return is £18.70.
The second ticket is valid either via Spalding or via Newark, but there is no train via Spalding on the way back. You could faff around with excesses by buying the cheaper Route Via Spalding fare, but the saving is less than £1 so easier to not bother.
Sample itinerary on the way back:
Gainsborough Lea Road 1738 - Lincoln Central 1803
Lincoln Central 1834 - Newark Castle 1858
Newark North Gate 1917 - Peterborough 1947
Peterborough 2016 - King's Cross 2147
The connection at Newark is a little on the tight side but perfectly doable for a normal person without mobility issues. Later itineraries are possible, including those that avoid the walk across Newark by taking a direct train from Lincoln to Newark North Gate.
Beware that as it is an FCC Only ticket between Peterborough and King's Cross, your friend must change at Peterborough if on an East Coast service.