So you are stating that your psu is so old that it does not have a single Sata power connection?
A regular Sata3 SSD uses a Sata power connection and a Sata data connection. Your mobo for sure has a Sata port, it'll have 4-8 of them.
An m.2 Sata ssd is the same as a standalone Sata ssd, the only difference being a m.2 plugs directly into the mobo, a regular ssd uses Sata cables.
Having a 2Tb hdd, you have Sata power, the older IDE power connections were discontinued long before drives of that size existed.
So not entirely sure why you feel an upgraded storage device is not compatible with either the psu or the mobo, since the necessary connectors already exist in your pc.
As far as slowdowns, it's doubtful you've filled up a 2Tb hdd, that's honestly a massive amount of storage, but it's also highly doubtful it's a 2Tb drive in a 2yr old pre-built, they mostly used 500Gb or 1Tb hdds as back then 2Tb drives were still considerably expensive.
The slowdowns are more than likely software based, multiple useless started services/processes, hdd that could use a good optimization, registry packed with orphans from uninstalled programs, caches bogged down with extensive temp files etc.
I'd start out with Ccleaner from piriform.com, analyze and run the cleaner a few times, use the default registry tool (yes save a copy) to clean that out, Defrag the hdd, use speccy (also from piriform.com) to check local temps (most should be between 30-40°C), do a general good dusting and cleaning of the pc (drag it outside and blow it out with compressed air can, especially fans and heatsinks).
Once cleaned, (inside and out), then check for speeds. You have a pc that's several years old, you might require a visit to the mobo's original site to install updated motherboard drivers, especially audio and Lan, since windows itself has had several major updates that can and will affect older, legacy drivers and create issues. For newest possible audio drivers you might need to go visit Realtek website if the audio drivers at the mobo website are dated.
Regular pc maintenance is a requirement for a healthy pc.