Hello there proposer,
For starters, that's the exact same situation I'm in now (how old are you guys btw?). Let me tell you it's easy to find a great one carat rock for $5K, and I'm not talking about the garbage ones either. I've been in the research phase for a few months now, and I'd be happy to share with you what I've learned so far. First off, in my area at least, I avoid local jewelers. Unless you really know the people well and trust them, or if they are an above average jeweler, odds are they're selling you mediocre stones at prices that rival the Ideal cuts of most online vendors. Personally, I like BlueNile.com for several reasons. Their selection is extensive, their settings are beautiful, their polices are great, and their customer service reps are more than helpful to answer any and all questions you may have for them. You'd do well to call them up and see what they can tell you, and they don't even pressure you into buying anything. They also come complete with either GIA or AGS certificates and a laquered wood ring box (sure to impress).
As far as the specifics of the diamond are concerned, I'm betting you're looking for the most shiny, sparkly, and firey diamond you can buy for your money, but also want a big rock to boot. Internet vendors make both wishes possible while sticking to your price range. Here's what I was told. If you want the best diamond that's noticable to the naked human eye, which is white and free of ugly inclusions, you should look for an SI-1 or better stone with an H or better color grade. I've seen a few SI-1's in person and belive me you can't spot these inclusions due to their tiny size and all the sparkly action going on in there. Colorwise, H is pretty white topside. There is an ever so faint hint of yellow on the side (just as there is a faint spec or two of inclusion on the side of an SI-1), but to hell with side viewing. I mean is she REALLY gonna look at the side and be critical when on the top it is white, clean, glittering like crazy? I think not. Especially if you get a six prong setting to further cover yourself, err, I mean, protect the diamond.
Don't expect perfection with a $5K budget. That's MORE than we should sanely spend on almost anything, and the small minute differences anything above that price bracket will buy is not worth it man! Pretty, sparkly, white, big, BAM! You're set. Microscopic perfection and gemologist level clarity can be damned when we're on a budget. PS, I've seen Hearts on Fire diamonds, which is one of the most popular incarnation of H&A diamonds, and let me tell you, to the naked eye, you aren't gonna notice much difference between that and a non brand name Ideal cut. He had one there for comparison, and he wanted FOUR THOUSAND dollars more for the HoF. That's just plain crazy. And don't go below .9 carats just to get super perfection either. The larger facets on a 1 carat rock more than cover minute differences in cut variances.
Well, that's my advice. Do what I did and call up as many online vendors as possible to get the most input and advice, and go to a local jeweler to see what clarity, carat, and color you're comfortable with (just try not to laugh when you see how their cut sucks). Also, if you're willing to drive, the Hearts on Fire website has a listing of jewelers that carry those diamonds, that way you can see what an Ideal cut diamond looks like (very pretty indeed). When it comes down to cruch time though, nothing beats online vendors like Blue Nile and the rest. Follow that advice, and your $5K budget will yield a rock the likes of which she has never seen before! Good luck man!