Most people use coolers for mash tuns, but living in an apartment myself I can tell that having the least possible equipment is a priority. I know some people use their secondary fermenters as mash tuns. How can one do that?
I converted my plastic fermenter into a mash tun/fermenter a few months ago and it works really well. Here are some pics showing how I did it:
In terms of heat loss, the heat loss from the water test earlier is a bit misleading because the water will lose temperature a lot faster than mash liquid. On my first all grain using this system I lost only 2 degrees over the course of an hour (I did have a blanket over the top of the mash/tun fermenter and then a belt tied around this). Use this for your mash and then simply wash and sanitise your mash tun while you boil, ready for it to be your fermenter.
Look into brew in a bag. You use your kettle for your mash tun. It sounds like you're nearly there since you already use a bag.
Okay, so the question is really about how to keep the temperature steady using a plastic bucket, so I'm reposting an alternate answer here, with my original response below.
Charlie Papazian's book also has an answer for this too. He suggests taking a large carboard box big enough to contain the bucket, and then placing the bucket inside the box and surrounding it with styrofoam. This can supposedly make a great insulated container.
He also suggests going to a fish market or seafood restaurant where they'll often have pre-fabricated styrofoam insulated boxes that can be super efficient at maintaining mash temps.
Hope this helps!
If you're brewing beer and you've never read Charlie Papazian's "The Complete Joy of Homebrewing", I suggest going out and buying a copy before you do anything else. It is the bible of homebrewing and has a section including instructions on how to create a mash tun out of a bucket system. Though others probably did it before him, the system is known as a "Zapap" lauter tun after Charlie "Pap"azian.
Basically, you just drill a million holes in the bottom of a fermentation bucket. Then, this bucket sits inside your bottling bucket. The bucket with the holes in it will sit perfectly inside connector for the spigot and leave a couple inches of space for a "false bottom". And there you have it, a mash tun.
It's that simple, and from what I've heard it works really well. The main drawback is that it won't maintain a steady temperature in the way that a converted cooler will, so you may need to add hot water from time to time throughout the mash. I've heard that if you make bigger beers using a lot of grain that the grains help to hold the heat better, so it's actually easier to maintain mash temperatures with big beers than with session beers, etc.
Here is a website with some pictures:
Eventually you'll want to get a converted cooler that will maintain a steady temperature over the course of an hour or more, but this is a good way to get into all grain brewing if you've been doing extract brewing and already have the equipment.
I do not recommend you do this. I am assuming your fermenting bucket is plastic.
You have to be super careful with your plastic fermenters. If you have to stick ANY implement into that bucket you risk scratching it. Scratches in plastic equal places to harbor bacteria and that means infection.
I know you want to save on room, but I honestly can't say that this is a good idea to do with your fragile plastic fermenting bucket.
I performed an experiment: I put some water at about 74 degrees C in the bucket, and then I covered the bucket with its cover, a large jacket and a blanket. I noticed that the part of the blanket on top of the fermenter was a bit hot, so I had some heat casualties from there. After one hour i measured the temperature and it had dropped 9 degrees C. It's not too good, but I suppose if I put one more blanket around, emphasising on the top of the fermenter, the temperature won't decrease by more than 3-4 degrees C, which is quite good efficiency.
I use my bottling bucket as a mash tun. I wrap it in foil insulation, and use some Brew in a Bag material (mesh curtains from IKEA) as a filter bag. I can mash 12 lbs of grain in it, and my old 5 gallon kettle lid fits snugly on top. I can keep a mash at 150 degrees for an hour. Yes, you can use a cooler, but if you already have a bottling bucket, it costs less than $20 for the materials, it already has a ball valve on it, and it's not another thing to store. You can throw the filter material in the washing machine after you rinse it out, too. I wouldn't worry about infection, as it only holds mash or beer for an hour or two, unlike a fermentation bucket that holds beer for weeks. If you rinse it out well and wash it you shouldn't have to worry about bacterial contamination - the mashing is pre - boil and the bottling is post - fermentation. But you should clean and sanitize the ball valve as much as you can.