I'm having the same problem, my version is 11.3.1.0 and haven't been able to get googleplay on my Kindle Fire HD 7" and it is already rooted. As you stated, phonesky isn't working because it's too old. You mentioned that your link was a fix for it but whenever I go to it, it's just another root tool and when I start it up, there isn't any options for googleplay, just rooting it. With an already rooted Kindle, I've already tried the steps before and have google framework installed and all that along with the vending apk. I can get the store to open in it's first stage and log in but after I log in, it closes. From there I cannot update it and am unclear as to what I can do in your link to get it updated. Thank you very much LinearEquation, you have been very helpful and it seems as though other people were smart enough to follow your advice previously.. I'm just a bit dull I guess :/
You can install Google Play on your 2013 Kindle Fire HD but it's for window shopping only. Any attempts to download from the play store will freeze the tablet and require a reboot. Really the play store serves no real purpose on the new KFHD 7, at the moment anyway. 1mobile and f DROID are a couple spots to get apks from.
Sent from my Nexus 7 Flo running Odex SinLess ROM 4.4.2 with ElementalX kernel using XDA Premium 4 mobile app