One way to make sure your gutter doesn't fall off the house is to install fascia hanger brackets. Installation is simple: Just hook the bracket under the front lip of the gutter, and then screw the other side of the bracket to the fascia. Leave the old spikes in place—a spike head looks better than a hole in the gutter.
If your shingles overhang your fascia by a few inches or you have steel roofing, buy the brackets with the screws built in (the type shown here). They cost more, but the head of the screw remains a couple of inches away from the fascia, making them a lot easier to install.