Depending on how your Logitech speakers are connected, you could connect the phono preamp directly to them (as your Logitech classifies as having a preamplifier and a power amplifier). In order to do this, they would need to be connected via digital to your computer (S/PDIF); then connect their front L/R (green TRS) to the phono preamp's line output. You'll switch to that input on the Logitech control panel, and use the Logitech system to control volume and other settings as well.
Further, as I'm remembering the Logitech having two digital inputs, you should be able to connect it in lieu of your HTIB as well, simply use the other digital input. On the remote you'll switch between the three to go between your turntable, whatever the HTIB was connected to, and your PC.
Now, this will not be the highest possible quality solution, but it doesn't require you to buy anything (unless you need cabling anywhere), and will work perfectly well. It also leaves a single speaker configuration in your room, which may make placement a lot easier.
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