The OP asked about doing the train so I was just informing them that it is a possible option for them to consider. Is it the easiest option then no it's not but it surely is an option. Just as an option would be WOW LAX-KEF-AMS or the direct DL flight to AMS. OP could also fly BA or VS with a change at LHR. or SAS via CPH

I'll leave it up to them to decide if it is sensible or not for them rather than just rule it out of hand which you appeared to have done. They said the trip is for work but that does not preclude a leisure element.

The distance between the Thameslink and Eurostar platforms is actually quite close - closer than some gate to gate treks at airports - I've done it a couple of times (though not after a 10 hour TATL) and it's all step free with lifts and escalators from the Thameslink Platform to the concourse level. Buying a train ticket is hardly difficult either - there is plenty of advice on here on how to do it and the OP can always ask for more info.