turkey3_scratch :
electro_neanderthal :
Let me repeat myself since you missed it the first time, with caps to help you see:
"Anything WITH an 80+ BRONZE rating or higher AND from a REPUTABLE company... (examples included)."
That is certainly not "get's anything".
Those reputable companies all have 80+ Bronze units many consider to be poor power supplies. So if OP gets anything, provided he is following those conditions, OP can very well still end up with a power supply of poor value if OP does not look into a PSU that's more, err, fine-tailored to their needs.
Every reputable company has it's duds, and by now a lot of those duds are history. Not saying there still aren't any, or won't be in the future, but it's still far better to go by what I said than grabbing something from a cheap, no-name or non-reputable brand.
I mention going for reputable brands because some lesser companies fake their 80+ certification by plastering a sticker that says it provides more power than the internals were certified for. Like a 450W PSU certified for Bronze being advertized as a 500W unit certified for bronze.
Here's an official testing list if one wants to make sure, but it's rather long.
https://www.plugloadsolutions.com/80PlusPowerSuppliesDetail.aspx?id=0&type=2
Getting back on topic with the OP:
I still recommend whatever the video card specs say for a minimum recommended wattage. If that 345W unit is enough for whatever specs the video card demands, and it's not too old, then by all means I think it's better to keep it.
But - and the reason I recommend getting a new one - knowing the reputation of PSUs in pre-builts, and guessing the age of the OP's system, I'd say it's safer to get a new one, since it's probably not gonna last long - especially with the extra load.
Now, as for fitting inside that case... it's got an outer width of 6.7 inches. So a card that fits within 5.7 inches width should be fine... as for length... a card that came pre-installed with that case was the HD 8490, which is 6.1 inches long. But the best way to check is to open the case, use a ruler, and see how much space there is. Then just browse various cards and find their measurements in the specifications.