There's only one real solution: Games. Plural.
There is no single game that could help the system, as very few people are willing to spend the money required for a gaming device for just one game that interests them. People need to see an array of high-quality games that appeal to them. I just looked over the PS Vita games and found only one game that I would even be interested in playing, but that's a port of a game I've already played.
Unfortunately, low system sales will inhibit further game development, as developers will be averse to investing time and effort into the Vita if its sales don't pick up. But if there are few good games, sales will remain low. It's up to Sony to publish a diverse variety of high-quality games for their system if they want to gain ground in the handheld market.
This is where Nintendo has a clear advantage: they develop and publish a lot of games for their systems. And while their first and second-party games don't appeal to the entire market, they do appeal to a wide and varied audience. The 3DS sales are driven not only by it's strong library, but also Nintendo's reputation for creating great games for their systems - not to mention a plethora of third party developers who produce for Nintendo handhelds. They can sell systems based on reputation alone - consumers know the games are coming. Sony, while they do make some good games, make far fewer and appeal to a smaller player base.
Sony's strength in the console department is driven by strong third-party support, but they can't get that support for their handheld without stronger sales. What Sony needed to do for the Vita was to put some real development muscle behind it. Lay the foundation. Instead, they released a piece of hardware and expected the software to come. "If you build it, they will come" only works in the movies, and Sony has a long road ahead of them if they really want to get this thing off the ground.
