Chromecast may not work on the Tab S 10.1. (If it does - and the app is free, so try it - it's the best solution.) You can use an MHL adapter - but with Samsung you have to make sure it's for that particular tab - Samsung has a few different microUSB MHL setups, so an adapter that works on one Samsung device may not work on another one. (Thank you for understanding what "standard" means, Samsung - and ignoring it.)
Re: How do I mirror Galaxy TabS 10.1 to a LG smart TV
Agreed. DLNA is a standard that all TVs are *supposed* to use in order to facilitate sharing between them and other devices. The problem is that a lot of devices don't implement the entire standard, and some bits they do implement are customized to the point where they don't always work well with others. I suggest getting a remote from the Play store that specifically says it's for LG TVs.