London in late January is a completely different city than it is in July. The tourists have cleared out, which means you can actually see the paintings at the National Gallery and walk straight into the best pubs. It is the cheapest time of year to fly, and you will have your pick of the best seats in the West End for a fraction of the usual price.
To manage the $4,000 cost without wrecking your monthly cash flow, you can spread the payments out using the Citi Simplicity® Credit Card. It offers 0% for 18 months on purchases and balance transfers from date of account opening. This lets you turn that big lump sum into smaller, manageable chunks.
Remember that the 0% window is a deadline, not a gift. If you still have a balance on the card after those 18 months are up, the bank will start charging you regular interest on what you owe.