The Logitechs can't compete with the Infinity Primus. The reason the Logitechs are popular is because of their small size for fitting on a desk and all-in-one 2.1 package that you can walk out the store with, not because they have great audio fidelity in comparison to home audio speakers. It's physics. The P163s are more full range bookshelf speakers that regularly sell close to $200 with 6.5" drivers. These are entry level home audio speakers that you could build a home theater with. Like the A5s, they can be run without a sub and still sound good because of their frequency response. They have midbass output. So comparing the Logitech speakers to the Infinity Primus is a lot like comparing a moped to a motorcycle. LOL
I went from from Klipsch Promedia 2.1 to Cambridge Audio S30s (fairly comparable to the Infinity Primus) using an old AVR as the power source in my first upgrade to passive bookshelf speakers. I didn't have a sub at the time to use with them, but the difference was night and day in overall SQ. That's what you are improving upon here--getting away from tiny cheap drivers. The Infinity Primus probably come close to competing with the Audioengine A5s, and may even sound better depending on your particular tastes.
Then with the sub, that's an 8" driver on the Logitech that has 50w rms amplifier output. Compare to the 10" with 300w rms of the JBL. A 10" has more speaker surface and produce more bass than an 8", and then the amplifier power is significantly more. The JBL has strong usable output down to 30hz, which I'm sure the Logitech does not. Here's a review of the JBL: http://www.avhub.com.au/images/stories/australian-hifi/reviews/2011-05_to_12/2011-11/jbl_sub150p_subwoofer_review_lores.pdf
As I said, the weakest link is the Lepai. But if you read that other thread, then you know that people say it works wonderfully for $20. Then six months or a year from now if you want an upgrade, watch your local Craiglist for a used Denon, Onkyo, Yamaha, or Pioneer AVR. Probably easily find one for under $100 if you live anywhere with an active CL.