Hello
If even the splitter doesn't work, I am afraid you may need to buy a new headphone that has a single pin used for both audio output and input.
However, if you still want to stick to your Logitech headphone, you can try an external USB sound card.
You can find many varieties of such USB sound cards
here. Such sound cards are cheap and have two jacks that you can use to plug in your headphone's microphone and speaker pins in them.
Hope this helps.
Good Luck!!