If it was East End and without bags, I'd say Columbia Road flower market.
You could see whether there's a left luggage facility at Liverpool Street? Then they could leave bags there and you could head off into Billingsgate Market or to somewhere like Breakfast Club for brunch. It would then be an easy ride round to King's Cross on the tube from Liverpool Street.
Probably won't appeal to hangovers but the Museum of London is a good option on that side of London. As is the Geffrye Museum and Dennis Severs' House but you'd have to check opening times.
With hangovers, I'd leave the bags at St Pancras or King's Cross and head to Granary Square for brunch, then maybe walk along Regent's Canal - or head up to Camden for brunch and a wander down the canal back to King's Cross.
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