It sounds like you have great plans and you'll be sure to take in the romance of the city doing what you intend.
The London Eye has spectacular views on clear days as does the Sky Garden at 20 Fenchurch St, (colloquially The Walkie Talkie Building) as I'm sure The Shard would too but I've never been up.
Greenwich Park is my favourite of the London Royal Parks. It's a manageable size, hilly where a lot are flat, there's a lovely Henry Moore (no relation) sculpture, the Observatory is a gorgeous building and the views across the city from there are wonderful, you can straddle two hemispheres and if you're there late enough as dusk arrives you'll see the green laser they emit to mark 0° longitude which flies across the sky to the horizon. It's real close to the river too so there are plenty of nice bars, cafes and restaurants to spend the evening staring into water which always does it for me!
Do check out the Tower. There are few places in modern capital cities where such well preserved millennia old buildings still exist and it is special. From there walk over Tower Bridge to the south side and walk westward along the river past City Hall, take in the vast size of HMS Belfast moored on the river, then the Golden Hinde that's dry docked by Southwark Cathedral and straight into Borough Market where you'll find some of the best food in the city, both produce and ready to eat. Continue along the Southbank where there's often great buskers (I first heard Portico Quartet there when they were just Portico playing for pennies), a cool young and arty vibe exists down there, from skateboarding gangs and graffiti littered tunnels to the Tate Modern which is itself a stunning modern architectural treat even if you don't like the art within. Do remember though, all London Museums are free to enter (you only pay entry to special exhibitions) so even if you stay 10 minutes and figure it's not for you, you've lost nothing other than those 10 minutes and you may just find yourself breathtaken by a Picasso, a Kandinsky or a Rothko.
Continue west and you'll get to the Eye and have stunning views of Westminster and Parliament. Cross Westminster Bridge if you want to get closer and then walk north into Soho where the bars are kicking. Old Compton Street is the most gay friendly neighbourhood in town, Ronnie Scott's and the Soho Pizza Express where you'll see Joel Ross are just up the road. Soho Square and Golden Square are both really hip open spaces in the nice weather where the young arty set just hang shoulder to shoulder with the homeless and more edgy characters. And Berwick Street has a great street food market during the day and leads to the quintet of record shops in close proximity, Sounds of the Universe, Reckless, Sister ray, If Music & Phonica.
Whatever you do, it's hard not to find something in this city that's for you. If you need any further tips while you're here, feel free to DM me.
