So here is the fix. You can install all the Google apps from Android market without rooting Kindle Fire.
All you need to do is:
1. Go to Settings, on your Kindle Fire. Select Devices, and activate “Allow Installation of Application From Unknown Sources”.
2. Install ES File Explorer from Kindle App Store.
3. Download the required files from here and extract them.
4. Copy them from PC/Mac/Linux Machine to Kindle Fire by one of the following ways:
a. Email b. USB c. Dropbox
5. Once the individual files are copied to Kindle Fire, access the .apk files using ES File Explorer. Start with GoogleServicesFramework.apk. Reboot Kindle Fire and install all the other .apk files after that.
6. Google music is downgraded to work with Android 2.2 used in Kindle Fire. After installing Google Music, go to settings and activate your account. All the apps should work fine, except that the resolution of the icons are awful.