Well, Brian, congratulations on retirement! Hope I can provide some assistance.
Your budget is a sizeable one, and in my experience, there is a primary area to invest your money: the processor. RAM can be upgraded, different programs can be added / deleted, but the processor is forever. If you are intending to do photo and video editing, I recommend at least an Intel i5 or an AMD Phenom II X4. Both would be adequate for video / photo work.
As you have stated that you would be working with photo / video editing, a large hard-drive would be desireable (500 GB or larger - preferably larger). Given those two points, the only laptops which would suit your purpose would be large and rather heavy, making their portability of moderate value. They would also be rather pricey in comparison to a similarly-equipped desktop or all-in-one, in addition to usually having much smaller hard-drives.
If not a laptop, the then question becomes, desktop or all-in-one? I'm with you on the debate about wires all over the place - what a mess that can be. If you are a rather neat, tidy person, the all-in-one would probably be preferable: few wires to connect or set up - just plug it in and begin. Among the all-in-one manufacturers that I would recommend: Apple (an iMac), Lenovo (IdeaCentre A700 - 40244JU) and Hewlet Packard (200xt or 200t models). All offer Intel i-series processors (the HP and the iMac offer Intel i5, while the Lenovo offers an Intel i7), large Ram (6 GB +), and large hard-drives of 750 GB or more. Dedicated graphics are preferable, but not necessary (only the HP offers dedicated graphics as an option).
At this point the question becomes: do you know what software you want to use for your video / photo editing? Some software will only work on a Windows platform, some will only work on an Apple platform. If you have not made a decision on this point, you don't need to do so right away. There is a lot of superb software available for both Mac and Windows for photo / and video editing (though it may take a sizeable portion of your budget to get professional-level software).
The cheapest way to go about buying a computer to do what you want would be to just get a desktop tower, and buy any peripherals you need (like a monitor!). But if style and space are an issue, the all-in-one is probably more your ticket.
Best wishes for your retirement and computer search!