Hi Jaxx and
Theres isnt a best way, its either right or wrong
I would first of all check that the splitter is working ok by substituting it with another.
I assume that you have at least 2 provided ?
Have you got an extension lead that you could connect to a different socket, ideally the master on where the line comes in and use that on your modem in case theres an internal wiring fault ?
Problems like you describe are usually BT line faults but they are very reluctant to admit it or do anything about it unless you insist
Is it a new pc/modem installation or have you been using this setup for a time and its only just started happening ?
Is the pc ok for everything else apart from connecting to the net ?