Would it be convenient to get to any of the larger airports in different states like Pittsburgh or Detroit? For some random dates it looks like that could save you $300-500 in airfare. If you want to fly out from Ohio I think a lot of the $1k suggestions here are overly conservative, esp now that United and I think US have decreased their presence there. Looks like $1300 is more likely, increasing to around $1500 the closer you get to peak travel times (ie: summer) If you can get to Boston, DC or ORD easily then you have some other interesting options like flying to London with a stop over in Istanbul for $1,000 RT or a a 1-7 day stop over in Iceland for $950 RT or a quick stop over in Dublin on your way to London for $1,000 RT
I think around $15 for lunch and $25-30 for dinner would be a good ballpark unless you spend all your time in London. For breakfast we would usually just go to a grocery store and get a bakery item, banana and drink for about $5
If you are going out and about you may want to think about renting a car for the non-London part of your stay. Take the tube back to one of the airports and pick up your car there. They all tend to be on the outside of the city (except LCY) so you will avoid the city traffic that way. An automatic will cost more but we got one because I didn't want to deal with driving on the wrong side of the road AND using my left hand to shift.
For london itself you might want to look at the Premier Inn County Hall. There are cheaper places to stay but this one is right across the river from the Parliament\Big Ben and relatively easy walking distance to the major attractions. They are usually in the $160-250 range per night which is pretty darn good for downtown London
Depending on dates, where you choose to stay and what you choose to eat I think $3k is doable but much less than that might be a stretch

Good lord - our two week trip there cost waaaay less than that