Let me try and put a slightly different angle on it. I think you are getting some anxiety from the fact that you can't get a "good nights" sleep and this is bringing you down, making you more anxious and creating a bit of a vicious circle.
Historically, humans didn't go to sleep for a good 8 hours then wake up the next day refreshed. They took naps as and when they could, and that served them well for countless generations. Look at hot countries as an example where people shut down at the hottest part of the day, have a nap, then continue on. Its perfectly normal.
Also, as has been stated above, sleeping cycles can be a bit of a habit, but if you have a smart phone, there are apps that help with sleeping now such as
http://www.sleepcycle.com/
What this does is wake you up at the best time in your sleep cycle so that you wake up refreshed. I have no connection with this product, but it does help, and I know quite a few folk that its helped