Netgear R6350 1750ac Are Antennae detachable?

I realize I left out a piece of the puzzle. My Access Point is hooked wired by a run of Cat5e cable from my main router.

Thanks again. Your help is much appreciated. What would be a more powerful alternative to the R6350? Are we talking R6400 or 6700, or should I be looking for a higher AC router. I must say I can get a couple of bars in the den from the R6350 in the study. It's not enough for me to stream some 4k movies I've backed up from my personal collection to my 4k TV that average from 66 to 130Mbps. It was playing part of them then seizing up. I'm seizing up with the R6350 as AP, but that's another problem for a new thread. My previous solution was hardwiring the TV to the AP in the den using one of the ethernet ports. The TV only has 10/100 so I believe that was limiting my bitrate and stopping playing.

What I have is just an L shaped house in reverse. Study is at the top end of the L, and Den is at the end of L. Most of my router signal in the study is going out to the next door neighbors and the street. Would mounting it in the hallway outside the study as high as I can go before the antennae touch the ceiling be a better location? It would move it one wall closer.

You may be right about about the antenna problem possibly being a red herring. I'm not certain how well they work. I could never get the homemade ones I made to make a difference. 

The old routers were great as far as stability for my network. Couldn't ask for a better option reliability wise. I just need dual band and a more robust signal that can take me over the hump to get my highest bitrate objects playable.

I hadn't looked into just an extender as I need a very robust signal in the den in order to play the high bitrate 4k files on the Sony XBR 900E 4k TV. I admit the exenders and mesh networks are intriguing, but it takes me away from what I know.