Have you tried just elevating the laptop a little? Put two small blocks, 0.5 to 1.5 inch or so, in the back corners of the laptop to slant the whole machine. It will significantly increase convection cooling, and make the internal fans more efficient so they should be able to run more slowly. If you have to use a large thing, make sure only the corner is under the laptop - remember, you want to expose as much of the bottom of the laptop to air as possible. The slant gives the air a natural convection current even when the fans aren't running, and there's a lot of heat sink on the bottom, so this is useful even if you don't have vents there.
Adding 1/4 inch blocks at the front helps as well.
Also, if you run a dell there are fan utilities that may allow more fine-grained control over the fan settings. Some dell laptops
can control their fans to a great degree, but don't, and so it seems like you either get on full blast, or off, when a steady slow would be better for the fans and the laptop. You might be able to find something similar for other laptop manufacturers as well. You can adjust the temperature the fans come on at, and each temperature they go to the next speed level. They also monitor the graphics card temperature, and a few other bits that can help you decide when the fans actually need to run.
If you find that elevating the laptop works (and I've found very few laptops that didn't respond well to it) then you can get this Ikea laptop stand very inexpensively and cut large holes in the bottom to allow airflow to the vents and back of the laptop:
http://www.ikea.com/us/en/catalog/products/60150176/
You might be able to modify/hack/cut your current stands up to try this out as well.
And of course, in the meantime you can take a small desk fan and point it at the laptop.
Oh, also you can probably take the currently broken stand apart, or just hack away at the port, until you expose the connector, then hot glue or superglue it in a spot where it won't move again. It's already broken, why not see if you can fix it with a little force?