Having read all of the above comments on the Fire HD I decided to call Amazon due to having the exact same fault with my device.
I described the problem and had to go through trouble shooting with the member of staff I spoke to. Then after explaining that I have looked on the internet and told hem the exact fault that I have and the reason it happens the team member jumped straight in to offer me a 'discounted' refurb product(£71.20 plus shipping).
After telling the team member I needed to think about what to do as I am 1 year out of warrantee and I wanted to see what others were offered in the same circumstances.
I then discovered that there seems to be a standard protocol followed by Amazon with offering the discounted refurb products, the difference comes with the discounted end price varying massively.
I then did a little research on UK Trading Standards website and called Amazon back. I explained I felt that Amazon should give me a better price than the quoted £71.20 on the refurb as new products over here for HD are only £99.00.
Also if a company are aware of an existing manufacturing fault that means the product has a reduced shelf life(18 months- 2 years) then they have a moral obligation to take steps to fix the fault or inform the customer, in or out of warrantee.
Based upon this the team member then went to speak to his superior and they have now agreed to a free replacement of the product and free shipping- result!
I would urge others with the same issue to stick to your guns and know your rights- thanks to this I will remain a loyal Amazon customer and they will continue to get my money on the e-books and multiple other things I purchase on a regular basis- this is where they make the money anyway.
I hope this helps others.