One tetra down, more to go?

Question
I have a 10 gallon tank that I've had set up since October.  I started with one
small goldfish to establish the tank and then added 3 golden barbs and 3
blood-fin tetras in January.  They might not be ideal tank mates, but I keep
the tank temperature at about 72F.  Definitely warm for the goldfish, but it
keeps all of them in their comfort zone.  Aside from that, I do about a 25%
water change every two weeks and clean the tank once a month.  Just flake
food right now.

Anyway, my problem began just recently.  All the fish are looking fine, except
for the blood fins.  I already lost one and another seems to have fallen ill.  I
didn't really notice the symptoms on the first fish that died (I'm in the middle
of finals, so I'm away from them, working on projects), but the second seems
to be overly agitated, staying near the top of the tank and constantly on alert,
especially at night.  It's not schooling with the barbs or the other bloodfin
either and he's losing his color.  It has fin rot and I think it's back might be
getting a little curved too.  Everything seems to point to Neon Tetra Disease,
which I looked up.  I have since isolated the sick looking fish.  No idea how
they would have gotten it now, since I don't feed live food and they've been
comfortable in the tank for months.

So, I'm going to watch the sick bloodfin and see how it does, but I'm kind of
lost on what to do with the rest of the tank.  My last bloodfin is also losing
color, but looks normal otherwise and my other fish don't seem to be
affected (yet).  What can I do to make sure that I don't loose my whole tank?  
Plus, I only have one extra hospital tank and I'd hate to doom my last bloodfin
by putting him in with the sick bloodfin, but it may be necessary to not lose
the others...  Advice?

Answer
I can tell you right now that the Goldfish needs to be taken out of that tank and eventually will need a 55 gallon or larger to itself.
All Goldfish get 12 inches long and need large tanks to begin with so their growth isn't stunted.

The ideal temperature for a fresh water tank is 76-82 you might want to make it a bit warmer in there. You should never clean out the tank completely that could create problems. Just keep doing water changes.

If you want to be sure to keep the other fish alive the best you can do is pick up another tank which you can even use an air pump for temporarily and don't need another filter and then get a heater. Setting up another hospital tank is the route to go if you don't want to stick him in with the other.

There is no known cure for the Neon Tetra Disease there is also a false Neon Tetra Disease caused by bacteria but it's difficult to tell between the two with out lab equipment. Just do your best to separate them even if it means a new tank.