Chromecast's only connection to the TV (other than power if you power it with a USB jack on the TV) is HDMI, and that's audio and video only - no internet. The Chromecast device works only on wifi for its internet connection.
As far as speed, 2mbps is fast enough for 1080i video, and unless you moved your router to your neighbor's house, you should be capable of that much speed. 4k needs a little more, but even 5mbps is trivial for a wifi connection. Connecting a Chromecast to the router by Gigabit Ethernet won't make it accept the stream any faster - the stream isn't that fast. (It's like filling a bottle from your tap - will you get a quart of water into a gallon jug faster than into a quart container? The tap controls the fill speed. The same way, the stream controls the streaming speed, the container [the Chromecast] and the air [the connection] don't, unless the connection is slower than the stream - which means that you have a terrible wifi connection to your Chromecast.)
If the Chromeccast keeps trying to set up but can't (wrong password, or some other problem), plug it into a TV near the router and set it up. Then move it to where you'll be using it. It seems to need a better connection to initially connect and set up than to run normally after it's been set up. (Don't ask me, I didn't write the firmware or design the hardware, I just know I've had 2 of them with the same "problem" in different houses, so it's probably something to do with the device itself.)
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