when you use that single just for the internet, are the speeds fine with no problem?
now when you use that same line just for cable tv, is the picture fine for most channels?
now when you connect the main line to the splitter, and then have two other lines, one to the modem and the other to the tv, is this when the problem starts?
really cheap splitters can cause large signal lost, the 2-way splitter at radio shack that cost $9 has a 3.5db signal lost on each output. your $2 splitter probably caused a 7db loss which can cause a major degree in tv quality.
What will benefit you the most is probably a cable amplifier, some people wouldn't recommend an amplifier as it increases the signal but also increases the snr(signal to noise) ratio thus you might bring one thing in range while knocking another thing out of spec but amplifiers do work in some situations. In your case, if you have a good connection and tv quality when using the original line but not when it's splitted then a amplifier will most likely be a major benefit. If you're willing to spend $48 bucks on an amplifier, this can be used as a splitter. go to ebay and type cable amplifier look for a guy name LotsaSpotz, click on his 2 port amplifier and in that page there's a link that says "click here" for more info under descriptions. Read that whole page and you will be well informed and should be able to decide if it's right for you. goto dslreports and search for cable amplifier/splitter topics and read those to help, but I think the info this ebay seller has should be enough for you.