Hi , here is the manufacturer site listing supported graphics cards including some with noted HDMI support

http://global.shuttle.com/main/productsSupportList?productId=646

I could not ascertain the pci-e bus version of the subject computer but may guess is it is pci-e 2.0 .

Your HD5450 video card is pci-e 2.1 standard which may not be compatiable with pci-e 2.0 version bus or less due to higher spec power of the pci-e 2.1 bus. This could sometimes be resolved via BIOS firmware update. However maybe not.

I would recommend sourcing a manufacturer supported video card

http://www.amd.com/en-us/products/graphics/desktop/5000/5450

https://en.wikipedia.org/wiki/PCI_Express#PCI_Express_2.1

Your pet computer should be able to be device upgraded to support HDMI 

btw : some thoughts regarding your computer device upgrade 

after installing the hardware , if supported , start and enter BIOS setup and configure for " first seek pci/pci-e video" or some such rather than default first seek " onboard" - this will ensure system does not halt during booting

Ensure you have the video card driver&control panel setup file ( 200MB+) saved ready to run the installation file - windows may detect and try tio install default drivers, just cancel the process, then run the downloaded driver setup files

Try to first uninstall the onboard video driver via Device Manager > Display Adapters

Hope this helps