haha. thanks for the illustration. i am indeed confused.
the 'box that will do everything' you need is a DAC/amplifier combo. that's because your speakers are passive (i.e. that coaxial cable you use to power them), and most digital-PC audio DACs do not power passive speakers. you need the hi-fi element for that, to significantly step-up the amplification of the low-power signal that comes from your (i presume) 3.5mm pc outputs.
you could just get a nice stereo receiver, i.e. a nice amplifier for your speakers, and plug your computers 3.5mm output into them. but this will still be a crappy, noisy, unbalanced PC output. not ideal. that's why you need the DAC part too. this would plug into your PC via usb/firewire and convert/upsample that signal into something much better than your motherboard's onboard audio sound.
at this point for a happy result with those speakers, you're looking at a decent whack of money for a 'box to do everything'. it might be worth asking just why you have those passive bookshelf speakers, and is it worth sinking a couple hundred into a box to do everything if the speakers aren't worth a couple hundred in themselves. you could buy some active speakers and a DAC instead and cut out the need for the hi-fi amplifier to drive (bad) bookshelf speakers. if you had some really nice klipsch audio passive speakers or something, i would see the point. they have a very desirable sound. if they are hand-me-downs with nothing particularly special about them, i would probably consider that a lost cause and just buy new speakers (harmon kardon are a 'nice' brand when they come bundled with pc world desktops, but they're not really anything in terms of audio).
there are some very nice boxes that will do everything, 'integrated digital amplifiers' is what you should be searching for. many people with games consoles rather than PCs find they want to do the same thing as you. the bad news is you are looking for a 3-in-1 device: a DAC, an amplifier, and a headphone amplifier. you're looking at £400-500 to get into decent territory.
e.g. https://www.amazon.co.uk/NAD-3020-Digit … B00F0OMUGS
cambridge audio and marantz will offer nice solutions like this, too.
or, as i said above, if your current amplifier has inputs, you could buy a DAC that will route into your existing amplifier. i don't know how good your amp is and if it's worth sticking with it/investing in a DAC for it. same situ as with your speakers, above: are they worth this hybrid set-up or would it be better to start over? you could get a decent DAC for £200-300, say, and route its unbalanced RCA (red/white) outputs into your amp -- assuming it has them. many amplifier receivers will be multi-channel; hi-fis often consist of turntables, cds, tape decks, etc, after all. you would notice an improvement in sound quality but, again, it's subject to how good your amp/speakers are. a DAC alone will not miraculously transform what you hear.
i've recommended these before as simple, good standalone DACs:https://www.schiit.com/products/
or for a desktop DAC that could possibly route into your existing hi-fi amp:https://jdslabs.com/product/element-ii/