I am not familiar with a DSL splitter, as opposed to a DSL filter, but that might be a good idea. Is there a reason the DSL splitter couldn’t be installed somewhere down the line rather than at the NI?
First I would try putting multiple DSL filters in line with each other. I have seen this work. Make sure they are facing the right way because they are not bi-directional. Basically, if you have one facing the wrong way, you won’t hear anything downstream from it.
Second, I would try a different brand of DSL modem, if this is possible, because it worked for someone else. Sometimes it is better to have a solution than an explanation.
Third, I would make sure you have ADSL, not some other variety of DSL. Not all varieties of DSL are meant to share a phone line. Some need their own separate line. Make sure the kind of DSL you have is compatible with sharing a voice line, and that the DSL company knows you are sharing a voice line and has set up your DSL service appropriately for this. I mention this because you said you had the DSL service BEFORE the phone line.
The reason the noise disappears when you disconnect the DSL modem is clear enough. It is because the DSL modem stops chatting with central office. I think they call the DSL card in the central office a DSLAM or DSL access module.