Question: How can I keep ice from melting for as long as possible?
When trying to store food and drinks in a cooler, which would be the best way to stop the ice from melting for as long as possible in order to keep the food good and the drinks cold?
Answer: "Learning it the hard way, we now take block ice and put it on the bottom of the cooler. It may last 4-5 days in contrast to a day for ice cubes. It can be purchased at convenient stores, or you can make it yourself by filling empty milk 1/2 gal cartons and freezing them." - Don, About Camping Reader
Here's what former Camping Guide, David Sweet has to say about melting ice.

pack supplies already cooled.
What I do is freeze water bottles and as they thaw drink the water and yes I know your aren’t suppose to do that. Normally I can keep ice this way for 3 days. Make sure you keep the chest out of the direct sun also
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On a related note; often you might end up with food items floatin in some melted ice water. We picked up a second hand vacuum sealer and anything that we don’t get wett, we seal. At worst you have some floating bags, but everything stays dry, with not bits and pieces floating in the water.
Blocks of ice do work best and some places do sell it. If you have a “local” ice house nearby, their ice is much colder and lasts longer than anything you can make at home or buy at a convenience store.
Also, use a 5 day cooler. We have found that the Coleman coolers work the best w/o needing to spend big bucks on a marine or other cooler.
We do both for our boy scout troop and our personal camping and it works great and has for a few years now.