Here's my little bit of advice on starting to do headstands:
A good thing to do to get your head first used to having the increased blood flow is to just hang over the edge of your bed. They also have inversion tables if you are not able to do a head stand for some reason. I also saw many years ago a device where the pressure goes to your shoulders instead of your head and neck. I remember once being in a class where we stood on our heads without using our hands at all against a wall. That snapped my neck back into perfect alignment - felt amazing - but I don't think I would try that now.
Also, if you are just starting out it really helps to first do it against a wall. The shoulder stand against a wall is also considered to be a preparatory pose. Even if you just start the handstand position without raising your legs up it does you good. I would first though put my legs up against the a wall while lying down to get the blood flow from the feet relieved and then do the beginning of the headstand.
Make sure that you stretch out your neck nicely and your shoulders first. Also, you have to learn the exact place to put your head to get balance. Don't be surprised if this takes lots of attempts. It's like any skill. It will come fast to some and slow to others but eventually with enough trial and error and persistence most people can do it.
10 minutes is a great amount of time for a short meditation practice. Focusing on the sensations of the body or any other technique. That's good exercise to add for your consciousness. I love yoga time just because that's a time put aside where I can meditate and NOT do anything else.
But, if the exercise of meditating isn't appealing - perhaps listening to music or listening to a book on tape - maybe something motivational and uplifting. It's not really a good position for reading the written word.
If you have neck problems you can always do handstands. Those are great for building the upper body and take so much effort that those 10 minutes will go by lickity split - or if it's easy you can just go for a walk on your hands instead and get all the being upside down affects!