There are a a lot of models that can work for you, although you did not list any your existing system specs or what exactly is the issue with the system. Seems a bit of a waste to replace a computer because it has issues with streaming when it can be fixed easily. It's like buying a new car when you run out of gas.
Anything with a modern, Core series CPU will work, no need for dedicated graphics as the Intel HD graphics that are integrated nowadays are decent enough for HD playback. See what you can find in your budget that has an i3 or i5, with 8GB or more of memory. If it supports wireless AC, that's a plus (if you plan to stream over WiFi, that is).
Your right I should have offered more about what I was replacing. I have a HP Elite Dual Core still running Windows 7 Professional. My laptop, also an HP running Win 7 professional. Both are old, and the laptop is no longer fast enough to run Major League Baseball. My desktop will barely run one game as long as nothing else is running. They have been used daily and beat into the ground...I've got my moneys worth...
Reinstall the OS on what you have. That is the fault of many people who think their hardware is shot or worn down. A fresh install o fthe OS will make the systems run like new, and since they are both HP, they either came with some sort of Restore disc or some form of restore already on the system.