Should we just tell them we have a relationship or dismiss all the talk about us?
That's really a question the two of you have to answer for yourself. How you feel about the gossip may be different (maybe she hates it, maybe it doesn't bother her at all?) It's also somewhat dependent on the intensity of the relationship, if it's just casual and short term your response may be different than if it's something that's quickly becoming serious.
That said, I think there are some important considerations for you to keep in mind:
Firstly, gossip is "fun" for gossipers because it lets them feel like they have some sort of inside scoop. In other words, part of the value in gossiping comes from knowing something other people don't know. If you disclose your relationship, you take that value away. It's not much fun sharing a "secret" that everyone already knows. This applies somewhat to jokes, as well - they mainly have "power" because you respond to them. Often, if you disengage from responding to the jokes, and focus on your work, you may find that they stop. In general, people who gossip and tell jokes will have plenty of material in a typical office - they'll often just move on to whatever the next "fun" topic is once their current material has lost steam.
Another important consideration (that really applies to any substantial relationship, not just one in the workplace) is to make sure you're on the same page. If one of you expects to spend lots of time together at work once the secret is out - lunch, coffee breaks, etc - and the other one doesn't want to do that, there's going to be issues. Talking through how you'll handle your relationship in the workplace will be important. It sounds like you already did a little of that when you decided to keep things a secret, although it also sounds like that failed.
Finally, consider if your employer has any sort of official relationship policy. Often, and for very obvious reasons, there are rules within an organization about being in a personal relationship between someone with whom you have a structural relationship in the workplace - for instance, dating your boss, or being in a relationship with someone on the board, etc. Also, some organizations require that any substantial relationship is officially disclosed - usually done as a preventive measure to ensure that people in a relationship don't have the opportunity to abuse any power relationships.
There's also the obvious point to be made - having a workplace romance can mean things get really awkward or difficult if the relationship ends poorly. This is definitely something to keep in mind, and it's worth the effort to consciously decide if you're really interested in the risk of a messy breakup impacting your career. Some people are sensitive enough about career impacts that it's just not worthwhile, while others may want to pursue the person they're interested in, no matter the workplace impact. Again, as above, it's not really our place to make that decision for you, but it's absolutely worth considering.