I had this exact problem. The data rate would drop from 16 Mbit/sec to 1 - 2 Mbit/sec when an external monitor was connected. It seems to be that the signal going out through the HDMI port interferes with the WiFi.
I was able to improve the data rate so that it was pretty much unaffected by the external monitor connection. You can improve it by rearranging the cable so it leads straight to the side away from the T100 and by using the right cable.
A Mediabrdige HDMI to Micro-HDMI cable works the best for me. Using an HDMI to micro-HDMI adapter with a regular HDMI cable worked the worst.
I describe what I did in more detail below.
The upload rate was never affected by the cable, only the download rate.
Googling showed me that there are people with laptops from various manufacturers who also find the data rate to be affected by plugging in an HDMI cable.
I eventually found that the data transfer rate depended on what cable I used and how it was arranged.
Worst transfer rate was with a straight hdmi cable and an hdmi to microhdmi adapter at the end. I noticed that if I put my cellphone on this cable then the wifi data rate of my cellphone was affected too.
A Cable Matters hdmi to microhdmi cable was better, but not perfect. If part of the cable was draped over the T100 then the data rate slowed quite a bit. If the cable was arranged so it led straight away from the t100 then the data rate was faster, but still not as fast as without it connected. Data rate was maybe a little better with the charger plugged in. My theory is that this keeps interference from going in the microusb port.
The following cable that I got from Amazon was much better.
Mediabridge FLEX Series Micro-HDMI to HDMI Cable (6 Feet) - High-Speed Supports Ethernet, 3D and Audio Return [Newest Standard]
The cable gave me a download data rate of about 18 Mbits/s as tested on speedtest.net, which was about the same as I got without the hdmi cable connected.
When copying large files via network neighborhood to or from another pc then I get a transfer rate of 5-6 MBytes/s. This was unaffected by whether or not the cable was connected, so for me, I consider the problem resolved.
The data rate from another pc using the Unitek wired ethernet adapter was 10 MBytes/sec to the T100 and 20 MBytes/sec from the T100. I did not try moving the T100 closer to my WiFi router so I do not know if the wifi data rate can reach this level.
UNITEK USB3.0 3 ports HUB with 1 RJ45 10/100/1000 Gigabit Ethernet LAN Wired Network Adapter with LED indicator for Notebooks, Ultrabooks and tablets with USB ports, USB bus powered
I read somewhere on line where someone thought the Unitek adapter was slow and that other ethernet adapters should be faster than the unitek one.
When I first had the problem I contacted Asus. They did not offer any helpful advice. They just offered to give me an rma number to get it repaired.
I use it mostly for media viewing like Netflix and it is definitely much better than my old Asus TF700,very smooth operation and better wifi connection. I was worried about the screen resolution being lower than the 1900x1200 of the TF700, but it seems quite clear. You can set the T100 to use the external monitor at its resolution, 1900 x 1200 in my case, along with the internal display, just like for a regular laptop.
I also tried plugging in an Edimax USB3.0 wifi adapter. Its data rate was unaffected by all the cable switching but for some reason the upload data rate using it was much slower than with the builtin wifi.