Try in a terminal:
sudo cp /lib/firmware/brcm/brcmfmac43430-sdio.AP6212.txt /lib/firmware/brcm/brcmfmac43430a0-sdio.txt
Then reboot.
If the wifi specs for this pc are listed correctly in the link you provided, ubuntu is looking for the brcmfmac43430a0-sdio.txt, but for what ever reason it is in the firmware folder as brcmfmac43430-sidio.AP6212.txt. This command will simply make a renamed copy of that file.
It's hard to say if this will work without the output of:
dmesg | grep brcm
(maybe add it to the question)
Also you may want to check linuxium's isorespin.sh. It can be used to create .iso's of ubuntu and many other distributions specifically for these atom single board computers, including wifi and audio.
Edit
brcmfmac43430-sidio.AP6212.txt is installed by isorespin and does not come preinstalled on ubuntu. It could probably be downloaded from somewhere, however isorespin adds a lot of other functionality like audio and some specific kernel parameters. A prespun version of ubuntu is available here. However I would recommend creating a lubuntu or xfce version for your mini pc because it has very limited memory.
