The only way that I could see the existing equipment being of any use would be if were to run a lead from your netbook's audio socket to the input of the speaker system (which, I assume, has its own amplifier built into it). However you'd face three problems:
1. Unless the speaker system has a 3.5mm audio input, intended for connection to the headphone socket on audio equipment, you'd need an adapter lead to be able to connect to the system's phono inputs (or whatever else it's got) ;
2. You'd then need to take caer to balance the output level from your netbook to match the required input level of the speaker system ; and
3. Unless you bought some sort of switching system, you'd need to keep unplugging different leads to switch between the input from your TV set and that from your netbook.
It would make far greater sense to either buy some conventional (amplified) computer speakers or, if your netbook supports Bluetooth, to buy a Bluetooth speaker.