The "business case" idea is the best advice, I believe.
Estimate how many hours of productivity you lose per day. Even a half hour is a lot. If it's reasonably more, then your case is even better.
For instance, measure how long it takes to compile your projects /start up/whatever. Have things you can objectively point to. Compare that to your target machine.
Multiply the lost hours per day by your hourly rate (the bookkeeping rate, which will be higher than your actual salary, or use your salary+benefits if you cannot find it. Ask your producer, PM or boss). This will give the daily loss in dollars (or euros, pounds, rupees, whatever) per day. Divide this amount into the cost of a new box. That will be the number of days it will take for them to recoup the costs of a new machine over keeping the old machine.
Show those numbers to whomever does procurement. Understanding the business case is the way to move anything in a corporation.
e.g. if you're losing $20 a day and a new box is $2000, it will take 100 work days for them to see a return on it versus not doing it. That half a working year is probably worth it, since you'll probably be there at least another year.
Also, point out that productivity gains are exponential, i.e. productivity builds on previous success, and slow initial productivity leads to lower rate of productive gains later on.
Also, say something like "I am afraid that my reputation with my colleagues might be suffering, because the machine crawls so badly". This will directly address any grumbling about your productivity so far.
Under no circumstances get your own machine.
As for your for working from home, I'm not sure if you received documented permission to work on your own machine. If you did, you can tell them you have been working from home because your home box is more efficient. If you did not, then do not bring it up.