This is a typical problem with HTIB systems. Those plugs on the speaker cables are proprietary, not anything “off the shelf” that would allow the user to swap out with equipment from other manufacturers. It’s possible that some other manufacturers use the same plugs, but the likelihood of anyone here knowing which ones do is remote, as the enthusiasts here generally have little regard for these low-end pre-packaged systems. Problems like you’re experiencing are only one of the reasons.
You might be able to get a new HTIB system from a different brand and simply splice the plug ends it comes with onto your current speakers. That’s assuming the impedance if the old speakers is compatible with the amplifiers in the new system.
Which is another reason we don’t like these things. Basically they are self-contained systems that are not designed to have their parts “mixed and matched” with the products of other manufacturers, as you have with a standard-component home theater system. True, standard systems are much more complicated to set up and operate, which is possibly one reason why you went with a simple all-in-one system. But the trade-off for ease of set-up is strictly-limited flexibility.
Regards,
Wayne A. Pflughaupt