Day 3
This day doesn't really require you to visit these destinations at certain times, the times are mainly recommendations here. If you want to stay longer or less time at a place it is up to you.
8:00 - Wake up
Breakfest at the Premier Inn is £11 (around $22) per person, so you will have a budget around $45 in order to eat. It is a self-serve breakfest and doesn't have limits on how much you can eat, so have as much as possible before going out for the day.
Menu: https://www.premierinn.com/content/dam/global/restaurants/Premierinn/premier-inn-breakfast.pdf
After your breakfest, go to South Kensington using the District Line from Ealing Broadway.

10:00 - National History Museum
The National History Museum has many giant exhibitations, covering many different categories of history and science, the building itself is 3 stories tall. It should take you at least a couple hours to go through it all.
14:00 - Buckingham Palace
After you are done with the National History Museum, hop on either the District or Circle lines and go to St James' Park, in which you will then walk to Buckingham Palace. You can take a picture here.
14:45 - Borough Market
After that, you can take the Jubilee line at Green Park to the London Bridge Station to reach the markets. The Borough Markets are known for their food, with some stores given prestigious awards. It has existed from at least 1014.
You have £100 budgetted to spend on food here.
Address: London SE1 9AL, United Kingdom

16:00 - Harrods
Harrods is the largest department store in the entirety of Europe (and one of the biggest in the world), spaning 7 floors, a 5 acre site with over 90,000 m^2 of selling space.
Address: 87-135 Brompton Road, London, GB SW1X 7XL
You have £300 Budgetted to spend on items here. (try they macarons they are nice).
20:00 - Dinner
After you finished with Harrods, come back to Ealing Broadway Station and eat at Tonkatsu, in which you have £30 alloted for the both of you. Be aware that the restraunt closes at 9:30, so if your taking this day's activities at your own pace you may not be able to get back on time, if that is the case, you can also go to the Popeyes (which closes at 12) or other restraunts nearby, as they should have similar costs.