As to not having any brackets to mount it, then maybe look for a nice cheap mid full size tower and it should work that will support a good PSU and room for you to put in your mobo with space to spare and look for HDD slots to accommodate you and for future expansion.
Well believe it or not I am still using the old Dell Tower I got in 2000, and this is what I have looking at front of the case. The very top slot was for my floppy and its removed, next 2 slots on the top (going down) is my 2 DVD's, third slot is empty and I modified that insert plate to hold a Hard Drive, and inside near the bottom I have 2 hard drives. I have the Corsair 650 Watt PSU and heaps of connectors to use so that is not a problem.
So previously I had external Hdd box like you had ok, and it kept bouncing around giving me delayed write failures in that USB deal. So I got a PCI-e Sata card and stuck it in my system Ok. I then use the very top slot where the floppy was in, and connected that with a extra long blue Sata lead and that worked. This one I was using it for my Flight Sim Games.
On the third empty DVD slot now modified I put in my Back up Hard drive and did the same and hooked with a extra long Blue Sata Lead to my PCI-e card (routing it thru the open back space and that worked no problem
The other two hidden Hard drives is as follows: Windows 7 Ultimate, and the bottom is for my Windows XP. The DVD's and the 2 hidden Hard Drives are hooked to the mobo SATA connectors and my problem disappeared. Been running OK since. I can easily also switch the DVD Sata Leads and take them out of the Mobo Sata and put those on the SATA Card and use my other 2 HDD's into the Mobo SATA slots. Either way both will work fine.
On my Bios (Gigabyte) when I start it up I hit the F12 Key, then it shows a list and I go down to Hard Drive hit enter, and then choose which Hard Drive then hit enter, that I want to boot into and viola it comes up.
Result is no more delayed write failures and I avoid files and security descriptors getting corrupted. So...... now I just got rid of that external hdd box.
So try it both ways of mine and of Mandarks suggestion. Today one needs to be innovative to stretch the dollar a mile longer.