TVs are usually not recognized devices in the Alexa app. Devices in this case usually refer to smartplugs, smart light bulbs, smart doorlocks, smart alarm/security systems, smart cameras. I have never seen a smart TV listed as a device in the Alexa app.
Whether your device is a TV, Blu-ray player, or something else, if it's not listed in the Alexa app under devices, you will most likely need to use an Alexa Skill to be able to control it by voice. If a skill is not available, that would indicate the manufacturer of the device is not supporting it for voice control. There are other ways to do it yourself, such as using IFTTT, Logitech Harmony, Samsung SmartThings, RMBridge, Raspberry Pi, Arduino, etc. Some of the latter items would involve some tech savvy and coding experience.
You mentioned that you already have SmartThings. And I assume that you already looked for an Alexa Skill for your TV and did not find one. Your next step would be to talk to the people at the SmartThings forum to see how they did it. I do not own or use SmartThings, but I do hear that there are some brilliant things that the SmartThings forum are doing with their devices. A Google search should help you locate the SmartThings forum quickly.