grip the head with a good pair of vice grips , using the the side of the vice grips. have the handle above the top of gutter, then tap the side of the tool with a hammer. this should get them out.
if not, get a drill bit the size of the spike and drill the head off each spike. then pull gutter off.
take the curved end of a flat bar,put the two teeth(?)on either side of the spike from the back side tap wth hammer till the head comes away from the face of the gutter,come to the front of the gutter using the same end of flatbar and hook it to the spike head tap it with your hammer and it will come out without hurting your gutter.darn,alot harder to tell how than it is to do
use a flatbar.take the curved end and put the twoteeth(?)on either side of the spike head from the inside of the gutter.tap the flatbar enough to get the head away from the face of the gutter come around to the front of it using the same end and hook it to the nailhead,hit the bar with your hammer and the spike will come out.simple eh?only problem you have now is when you put it back you will be putting the nail right back in the same hole-nothing for the nail to grip there are at least two ways around that that i can think of offhand if you are interested post back