I do not know your tablet but does the Bluetooth work when the dock is removed from the tablet and is it only the keyboard/touchpad that you are having troubles with? Can you "see" and"pair' with other BT devices with the dock removed e.g. a BT speaker?
If it doesn't work with any BT device, as you bought your tablet only a few months ago, I suggest that you verify the purchase date and if it still is within the warranty period as per the warranty documentation that came with the tablet, consult the documentation as to what you have to do to make a claim for a warranty repair or replacement.
If it is only having problems with the dock then still go for a warranty repair/replacement of the dock. If you wish to persevere to try and solve it yourself then the following is only a guess on my part and to prove it or not would most probably involve voiding the warranty. Just so you are aware of this if you wish to check it out.
I'm wondering if the BT module in the dock is either loose (as it appears from the parts list to be a separate replaceable item) or is faulty.
Here is a link to the parts list. Scroll down and click on pcb
Try opening the dock and reseating the module and see if this solves the problem. Ensure the dock is removed from the tablet when you do this.
Here is a link that shows how to open the dock on an Asus T100 so that you can gain access to the module.