Ok gentilmen , lets talk ice dams. First and foremost, what causes an ice dam to form anyway. Heat trapped in an attic, either from heat loss from the house or sun beating on the roof, cannot escape quickly and melts snow on the roof. This water drains to the bottom of the roof where it runs across a cold edge and freezes. The overhang of your roof (soffit area) is typically not over a heated area and tends to be very cold, much colder that the rest of the roof. Even with a well insulated ceiling and good ventilation ice dams will form if conditions are right. This is where the old saying,"an ounce of prevention is worth a pound of cure" rings true. Any time you have a heavy snow load on your roof followed by some moderate temps or sunny days, some natural melting is going to occur, especially if the ridge vents are buried in snow. Sun hitting the gable ends will heat up the attic a bit too. Many times the ice dam starts in the gutters then bridges over to the roof edge and starts it's way up. Once it starts, there is now way to stop it unless the source of water building it is taken away. The only way to deprive the ice dam of more water is to REMOVE THE SNOW FROM THE ROOF AND MAKE AN EXIT FOR ANY REMAINING WATER TO DRAIN. This is why it is so important to use a roof rake soon after a snowfall to remove several feet of snow from the edge of the roof. With the snow removed, you can see if a dam is starting and can address it quickly while it is small and easy to treat. Other things to consider may include, if you live in an area where your ridge vent commonly is covered in snow, add a gable end vent on both sides of the house and be sure the soffit vents are equipped with "proper vent panels" inside and that they are not blocked by insulation. Always use a minimum of 6 feet of Grace ice and water shield, or if you want a good insurance policy against leaks, cover the entire roof with Grace before shingling. This may cost a bit more up front, but pays for itself in the long run. Maximize your attic insulation. Take a good look at your gutters and consider lowering them if they are too close to the drip edge where ice dams begin. There is no one thing that will prevent ice dams, but if you look at all the preventions at your disposal, dealing with them will be a lot easier. Sorry for preaching guys, but I give this same speel to many customers every year when water is leaking in their homes and they are shelling out big bucks for repairs. Two years ago here in Maine, the huge snowfalls and subsiquint ice dam problems prompted homeowners insurance companies to send out letters telling policy holders that they would not cover a second loss if these precautions were not met.
Well, fixing the damage won't prevent it from happening again, so to fix this and prevent further damages from ice in the future, you will need to install some hardware. First, you are going to need to get out the ladder and , with some assistance, push the down spout back up and re-attach it to the elbow. I'd use extra fasteners, and possibly a short rubber connection hose to ensure a good tight seal. It is likely ice formation caused the metal to flex, creating a weakness in the fastener, so the rubber hose will help.
The gutter issue really comes down to weight. Installing some "Heavy Load Gutter Brackets" would help. For the distance of gutter and weight on it I'd put in two brackets with each in the middle of the roof jousts near the middle that are shown in the picture.
With all of these fixes, be sure your nails or screws get "purchase" or else it is all for nothing.
EDIT:Use the same fasteners you are using, just use more. Not a ton more, maybe just one or two. Don't use sheet metal screws, use wood screws. Sheet metal screws have smaller thread, wood screws are made for, well, wood. That will handle the horizontal movement. For the vertical sliding, either rivet the joints together(this option costs less, but more permanent, and is more invisible) OR buy a vinyl connector and put a fastener on the connector itself and on the downspout itself just above and just below the connector
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I had to score the top with a box cutter and then smash it with a hammer!