The thing about external PSU they pretty much come in one size for whatever computerthey are designed for and mixing one from another computer is just not smart. At this point I would seriously consider saving up and just building a new computer.
Here is a budget build that should do 1080P 60FPS without any issue. The mobo is showing a warning but it should have an updated BIOS already installed from the factory.
PCPartPicker Part List:
https://pcpartpicker.com/list/HbXGPn
CPU: Intel Core i5-9400F 2.9 GHz 6-Core Processor ($144.99 @ B&H)
Motherboard: MSI B360M BAZOOKA Micro ATX LGA1151 Motherboard ($54.99 @ Newegg)
Memory: Corsair Vengeance LPX 16 GB (2 x 8 GB) DDR4-2666 Memory ($52.99 @ Newegg)
Storage: Crucial MX500 500 GB 2.5" Solid State Drive ($64.89 @ Amazon)
Video Card: Zotac GeForce GTX 1650 SUPER 4 GB Twin Fan Video Card ($159.99 @ Amazon)
Case: Corsair SPEC-04 ATX Mid Tower Case ($48.96 @ Amazon)
Power Supply: Corsair CXM 550 W 80+ Bronze Certified Semi-modular ATX Power Supply ($69.98 @ Amazon)
Total: $596.79
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2019-12-07 02:37 EST-0500
This doesn't include a copy of windows, however you can download and install Windows and not activate it, you just wont have a few features as well as a water mark in the bottom corner until you buy a license. You could also save about 20 or 30 dollars and get a standard HDD but anymore having an SSD at least for the main system drive really makes computers much faster