As someone that has worked on heavy equipment most of my life I've only seen machines with that much water in the hydraulic system a few times. I understand what everyone is talking about when it comes to condensation and yes, that can put a little water into the system over time. However to get enough water to milk up the system like your describing there is most likely a leak somewhere in the system allowing rainwater, etc in, or more likely someone put oil into the system from a from a jug with water in it. That said while leaks allowing water in aren't uncommon, water getting manually from a contaminated jug is the most common way for it to get there in any major amount, at least in my experience.
As far as getting the water out of the system there are filter cartridges available that are designed to draw the water out of the oil. The problem is these are usually designed to be used on a filter buggy and not on the machines system, or if used with the machines system have to be specially plumbed in so as to not damage the system when the filters become clogged/waterlogged, etc. Too these type filters are usually expensive to replace so depending on how much water is in the system as to how many of them you'll need to get all of the water out. In other words even if you went this route and put a filter buggy on the system or went to the expense of hardpiping a filter housing inline, you'd still need to get as much of the water out before hand as possible. The easiest way to do that is by changing the oil several times.
As bad as I know you hate to hear it, a 15 gallon system is actually pretty small as far as hydraulic systems go so your best bet would be to simply change the oil out a few times and take things from there. The way to do it is to work the machine and get the system as hot as possible each time before draining it. I know everyone says that getting the system hot will burn the water out and to a degree that is true but once it becomes emulsified/bonded with the oil your never going to get all of it out that way. I've seen too many people try it and fail to tell you it can be done, because it just ain't gonna happen like that unless you were to run that machine 24/7 for longer than you want to think about it. When the mix turns gray the water and oil are emulsifed/mixed together and getting them to seperate by just running the machine is pretty much hopeless. What will happen is that by running it enough to get it hot you'll get everything mixed up really good so that there will be less free water in the system to get missed when you do drain it. Too with hot oil you'll simply get a more complete drain than you will with cold oil just like when changing the oil in your engine. Also before you drain put every cylinder on the machine in the position where it holds the least amount of oil. In other words suck in every cylinder possible as the rod end holds less oil than the other end. If possible on something like a backhoe don't be afraid to put the boom up in the air which will push oil out of the top of the cyliner and the when the system is drained lower the cylinder and push the oil out of the bottom of the cylinder also. True this won't work on all equipment, and not on alot of the newer stuff with all of the "safety" mechanisms, but it will work on most older equipment. OK, once the oil is drained, if it is possible to spin the engine over without starting do so. This will often cause the pump to push fluid that would otherwise stay in the system back into the tank where it will drain...again DO NOT START THE ENGINE, JUST SPIN IT OVER WITHOUT STARTING....Once you get all of the oil possible out fill the system with fresh oil and immediately run the machine enough to get the hydraulic system hot again. This should allow time for much of the emulsified water and oil left in the system to make it's way back to the tank. So, run the machine for a little bit and then shut it down and drain it again just like I described before, but don't let it set for any length of time before you drain it, just like before drain it hot.
After doing this a time or two the system should be nearly free of the emulsified water/oil and any remaining water left will be of a sufficiently small volume that it will be able to 'burn off' when the system gets hot. Too it wouldn't hurt to try the tank drain each morning for a week or so to get rid of any free water that may have settled out overnight.
I know even with a 15 gallon system that it can be quite expensive to do this several times but if you've ever seen what water in a hydraulic system can do to the pump, the valves, and other system parts you'll understand that the cost of the oil is a small price to pay to keep from paying the big bucks needed when the pump fails and fills the system with debris....then your really talking BIG money to clean the system out.....Been there, done that, for more customers than I care to think about.....
I know change the oil wasn't what you wanted to hear but with smaller systems it's usually alot cheaper than the alternatives which are usually quite expensive. Below I've included a couple of links to Machinery Lubrications website and two articles on water in hydraulic oil. This might give you a little better idea of the damage that can be done as well as introduce you to their site which has alot of good info on the ways and products to keep this problem from happening again.
Damages from water in oil
Getting water out of the oil
