Try Expedia on your end. Make a matrix of all the airports you can fly out of and into. (On the UK side, include Glasgow as well as Edinburgh. It is a fairly easy one hour a way by train.) Make a list of possible outbound and in bound dates. By changing your dates by a day, sometimes you save hundreds of dollars. Try all of the different combos.

Your difficulty is that you are so close to your departure date.

I am new to this travel site but am a regular and Local Expert for Edinburgh on another well know travel site.
You have had some great suggestions so far.
The National Museum of Scotland is a great place (just a pity next door is going through a refurbishment) and it does mnay free tours so it is well worth visiting.
http://www.nms.ac.uk/nationalmuseumhomepage.aspx
There are lots of other small but fun museums in edinburgh as well,The Museum of childhood has been mentioned but further down the royal Mile is the peoples stort and the Museum of Edinburgh,these are small museums but buildings themselves are worth the visit.You could spend a week just walking the Royal mile and this website should give you lots of info.have fun.
http://www.edinburgh-royalmile.com/

I flew on Easy Jet from London to Edinburgh.

I am going to Edinburgh in a couple of weeks for two days, these tips are great, have another question, what will the weather be like, looks like it will be chilly, any recomendations on what to pack.

likely to be wet and windy ,today it is quite wild here and as I look out my office windown the rain is sheeting down and the sky grey grey grey.
best thing to do is to wear several layers rather than heavy outer coat.fleece ,body warmer and a waterproof/windproof outer layer is the way to go.
we do get some nice days late October /early November but early mornings and late evening can be cold,frost is a possibility but the wind is the thing that really makes it feel cold and we get a lot of wind in the city.
use a hat rather than an umberella.

A few hours later - sunny and windy. Daytime temperatures in the last few days have varied from c.6-c.13 degrees C.

yep,I am sitting having my lunch looking out the office window and it is a completely diffferent country from 5 hours ago,might just finish my sandwich and go for a walk.

er77, any news ? Have you managed to get a flight yet ? I assume you are not here now ?