Windows 10 added the capability to cast over the network to a streaming device. Some smart TVs can act as receivers, but since yours is not, you'll have to get a supported streaming device like a Roku or Amazon Fire Stick. Windows uses Miracast, so you'll need to check if receiving device is compatible. Then it's just a matter of plugging that device into your TV, and setting your TV to receive input from it.
Edit: The April 2018 Windows update may have moved the location of this feature. It's now in settings -> display -> connect to wireless display. Not settings -> devices as in the linked tutorial. (I really hate how Microsoft keeps moving these things around, making most of the guides you find online useless. I mention this so anyone reading this in the future knows not to give up if Microsoft decides to move the setting again in a future update.)
https://www.intertech.com/Blog/windows-10-tips-streaming-and-cast-to-device/
https://www.wi-fi.org/discover-wi-fi/miracast
https://en.wikipedia.org/wiki/Miracast#Devices
Google Chrome (the browser) also has the ability to cast your browser or desktop to a Goggle Chromecast.
https://www.howtogeek.com/199565/mirror-anything-from-your-computer-to-your-tv-using-google-chromecast/
While these work, the image quality will not be as good as a straight HDMI connection. Casting works by using the GPU to first turn your desktop into an h.264 video stream, then transmitting that over the network to the receiving device, which plays it as if it were playing a streamed video. So compression artifacts and network dropouts (especially over WiFi) are inevitable.