bose headphones are not too good compared with what else is on the market for the same prices so you may not notice improvements as much or need as good of equipment. they also likely will not benefit from an amp much and only a bit from a dac. wireless speakers also are not going to benefit from an amp but will benefit slightly from a dac. so while you need an amplifier on a unit if you bought something you're not going to need anything with high output for them. the dac portion if its better than onboard you may notice but mostly only if you have very bad onboard right now otherwise there will not be significant improvement noticed on such equipment as you have.
now, could you benefit from either a soundcard or dac+amp unit? you might notice a bit more clarity or perhaps slightly more bass on the headphones but thats about it. the speakers and headphones you currently have really arent high quality enough to warrant spending much money on anything to drive them.
at one point in time i owned a pair of bose similar to those and i thought they were half decent (compared to the earbuds i used before) however when you compare them to even budget level hifi headphones (like the m50s i own now) it really is a night and day difference and i certainly will not go back to bose whom only rely on their name to sell products which are inferior in build quality and sound. build quality is crud on bose and the cable is going to break off eventually as happened to mine though this is not unexpected given how frail the wires are.
if you do want to buy something to upgrade what you have, perhaps a xonar dg internal soundcard if you have poor quality onboard audio. if you need an external the e10k would work however certainly is quite overkill for what you have.