score:12
Since the flight was operated by KLM, the flight is covered by EU Flight Compensation Regulation 261/2004. Relevant for the calculation of the compensation is your delay at the final airport. From what you write in the question and assuming that the delay was the airline's fault, you are entitled to a compensation of 400€. The distance from Moscow to London is about 2500km and the delay in London was obviously more than 3 hours, since you had to spend a night in Amsterdam.
You may do as lastminute.com is suggesting and mandate AirHelp to claim the compensation. AirHelp will charge at least 100€ for this service. You may also continue with the claim yourself, depending on where you live, perhaps get free help from public consumer protection organizations in the country you are living, go to any of the other AirHelp-like service providers or order a lawyer to present your case (which may actually be cheaper than using AirHelp or other service providers).
There is no period set in the EU regulation how long a claim is valid. Your claim will lapse just as any other civil claim based on the periods defined in national legislation. 5 months should in any European jurisdiction be with large margins within the limits.
Upvote:2
UPDATE: We decided to submit a claim for compensation through AirHelp. The process was quite slow as they needed to take legal action, but eventually, we got the €400 per person we were entitled to, minus any fees.
Upvote:7
IANAL/IANYL. That said, the ECJ (and some higher national courts) have recently been shedding light on how this directive is to be interpreted. As this question makes clear, it's time lost at final destination that determines eligibility for compensation. Since you missed your connection and were twelve hours late, you're entitled to compensation, assuming your flight is covered.
Note this directive cuts both ways: since the passenger in the linked question was severely delayed on his first leg but didn't miss his/her connection because of the long layover, (s)he was on time at the final destination, and thus isn't eligible for compensation.
As your flight was operated by KLM, a European operator, it's covered. In the UK you generally have six years to pursue breach-of-contract-type remedies, and I'd imagine other EU nations have comparable timescales for seeking this type of redress. I'd contact the airline and ask for compensation according to the directive. Be ready to use the court system if it proves necessary.