The Kindle Fire HD has a
micro-HDMI (micro-D connector), which is different than a mini connector.
I don't have a Fire, but my two Acer Iconia tablets have a micro-HDMI port. It works fantastic! We use it to stream Amazon Instant Videos from tablet to TV.
The first cable I bought didn't work well because it disconnected whenever the tablet was moved. I think the connection was loose at the device. Then I purchased the
Mediabridge Flex Series - High Speed Micro-HDMI to HDMI Cable, which works perfectly. I've never had a disconnection problem with that one.