With those loan rates, I would use it to kill the debt. With this 10K and then the 1K a month you are already doing, you will be able to kill all of the high interest loans quickly. Then you can start investing.
The biggest thing is to recognize if you are suffering from any irrational psychology that could be clouding your judgement, for instance 'mental accounting'
http://www.investopedia.com/university/behavioral_finance/behavioral5.aspThe short version is that money is money. If this money came from a different source (say you just took home 10k more than you actually do), what would you use it for? That answer shouldn't be any different than your answer for the 10k as a windfall. Money is money, it doesn't matter where it comes from.
Consider these questions: If you found a crisp $10 bill on the street, what would you do with it? What about $100? $1000? The answer to all these questions should be the same as your 10k answer. Since you effectively said 'we usually use all our extra money to pay down debt' this money shouldn't be used any differently.
Now, whether you decide to kill debt, invest, or a combination of the two, as long as that is the answer for all of these scenarios, you are fine, and that decision can be made off a number of factors. You can do the maximization strategy of getting the most out of every dollar, you may decide you want to be debt free and that's more important than the mathematically best option (investing after the 6% and higher loans are gone), it's your decision. Just make sure you don't have any of the irrational biases in that article affecting your decision.