Danny, ah, sweet London... I think it's romantic no matter what you do :)
The super tourist in me says to do the private champagne experience on the London Eye where you get your own pod and have bubbly during your hour long ride (do it as the sun goes down to get a great view of the city in both day and night).
If you love the idea for a view, there are plenty of options instead of The Shard (which is, from what I've heard, overprice and not as great quality). These include Duck & Waffle in the City -- you'll be in the heart of the modern buildings, but it's a beautiful view from the ~40th floor; Barbecoa with a view of Saint Paul's Cathedral; or Oxo Tower or the Tate Modern Cafe for views of the Southbank and Victoria Embankment.
Personally, the most romantic thing I'd want to do would be to have dinner at somewhere intimate like La Bodega Negra and then move onto a booth in Nightjar. London has great views, but it's "underground" scene (literally) is what I prefer.
Hope you have a great time!
If you are interested in films London does have a few unique cinemas, The Electric Cinema in Notting Hill is one of the oldest cinemas. You can also watch a movie under the stars, London’s outdoor cinema season usually runs from late spring to September, with more screens popping up every summer. Among your choices are the Rooftop Film Club in four urban locations across town, Luna Cinema, which tends to present evening screenings in pretty parks and squares, and Dalston Roof Park where you pay £5 membership and can see films for free. And I would visit Borough Market for a unique food experience .
check out Sushisamba - they have a great restaurant and cocktail bar, if your on a budget it is pricy but worth it to go up for a cocktail! another option is Vertigo 42 which is a champagne bar on top of tower 42. My fiance worked here and it was a nice view of the city if thats what your looking for. it does get busy though so try and get up there early enough to get a good seat.
Also forgot to mention Madison Restaurant is a must - it over looks Saint Paul's Cathedral and is stunning. Cost nothing to go up and have a drink and even dinner if you want to try out there resturant
Romantic stroll along Broadway Marketis beautiful, especially via Regents Canal.
Duck & Waffleis extravagant, but worth it especially for the setting!
For something quirky though, Drink, Shop & Do is a winner for quirky romance!
Go see show or concert at the last minute. The English have a laudable and very civilized policy of allowing ticket owners to return their tickets if they can't make the show (I wish we had that here). This means that 1 hour prior to a show (or some places like the National Theatre when the box office opens) they often have seats available, usually at a discount. I've seen great symphony concerts this way, and plenty of shows. Just do your recon work ahead of time, speak to the box office workers and ask when the best time to score returned seats is for that theater, and go for it. Obviously you don't want to see something depressing or dour so again do your homework on what shows you would like to see. Spontaneity is always a big win - but have a back-up plan like drinks in a funky little wine bar in your back pocket just in case.
London's most romantic restaurant... Clos Maggiore