This one is a little tricky, If you had done sudo apt install kile --install-suggests
you would have installed a great deal more packages, and through many of the 'suggested' packages you would have arrived at 'texlive-latex-extra' which includes the file 'arydshln.sty'
I'm generally not a fan of 'install-suggests' as sometimes it installs far more software than I want.
For example sudo apt install kile
on my system will install 197 packages, loading 131 MB of archives
sudo apt install kile --install-suggests
will install 1641 packages, pulling 4,352 MB of archives.
In this specific case, I believe that you would be best served by
sudo apt install texlive-xetex
This will pull in the package containing 'arydshln.sty', and about 20 others some of which apparently are wanted in your specific application.
Further information and investigation
You can look at the packages suggested for installation using the command
apt show kile
and at https://packages.ubuntu.com/ you can search for the packages mentioned to see what they require as well. I used this webpage to search for the file 'arydshln.sty' to find what package it was contained in, and apt-cache rdepends texlive-xetex-extra
to find eventually, the chain that led to one of the suggested packages for 'kile'
arydshln.sty
on packages.ubuntu.com and you will gettexlive-latex-extra
as package name. This is how deb-packaging works. You can install only needed parts of texlive by using this way. – N0rbert Mar 12 '19 at 20:52