I've stayed 3x here: http://www.deneguesthouse.com/ The last time was 2 years ago and the prices are higher, of course, but perhaps contact directly and then compare what is stated vs the online booking engine (the online prices are way higher than I paid in '12 and I made the reservation by phone). They have a triple and a family room, probably better to book sooner than later if you want to try it. Walking from Princes St to the Dene you go right buy a hotel I've looked at and read good reviews here on RS is http://www.waltonhotel.com/ They also have a triple room listed.
For sightseeing, my favorite is http://www.edinburghmuseums.org.uk/Venues/The-People-s-Story .. I really like the video that plays in the tippy-top room; it covers life in the city from early 20th century to the 80s. Across the street you can see Greyfriars Bobby's bowl and pictures at http://www.edinburghmuseums.org.uk/Venues/Museum-of-Edinburgh
The National Library of Scotland has a couple special displays that will still be available in April: the great things that Scots have given the world http://www.nls.uk/exhibitions/whas-like-us and the book sculptures http://www.nls.uk/news/archive/2013/11/paper-sculptures
Not far from the Dene is http://www.cuckoosbakery.co.uk/ Even if you lodge elsewhere, stop in for at least one cupcake. I would not likely buy more than one (not cheap), but worth one stop on a trip, definitely.