As Kristi says, I'm afraid the bottom line is that it's really not worth the trouble.
For a start, a 2kW converter which sells for only $40 is going to be one of the cheap switched-mode types, probably giving a squarewave output instead of sine. If you try to run motors on that sort of unit, you'll have noisy operation and overheating. A decent transformer which will handle that sort of power is going to cost much more than $40.
Next you need to consider the fact that larger motors can draw a considerable surge of power when they start (even when running normally you can sometimes notice lights dim momentarily when a heavy motor starts up). That means that you need quite a hefty transformer for appliances which have a large motor. For example, a true 2000 VA transformer would normally be suitable for motors up to about 1 horsepower (746 watts).
On top of that, there is then the problem of different supply frequencies. The U.S. uses 60 Hertz (cycles per second), whereas the U.K. uses 50Hz. Transformers convert only the voltage, not the frequency. The lower frequency is not generally a problem for the smaller types of motors in food mixers, blenders, and so on, but it becomes much more significant with the more powerful types of motor found in refrigerators, washers, etc. Using these on the lower frequency reduces efficiency and can lead to overheating. Note also that the timers on some washers are also synchronized to the supply, so running a 60Hz washer on 50Hz might also result in the cycle times being extended.
The clothes dryer poses problems on top of those already mentioned. The typical American dryer has a heater which runs at 5kW or more, and the dryer is fed with both 120 and 240-volt power (240 for the heating element, 120 for the motor, timer, lights, etc.). You may have also noticed that in the U.S. some dryers are connected to the outlet with a 3-prong plug while others use a 4-prong plug. That's because the protective ground connections can be arranged different ways. All of this would have to be reconciled with U.K. power and any transformers, but that would still leave you with the frequency issue.
Apart from the physical size issue which Kristi mentioned, another point to bear in mind with the washer is that most American washers have both hot and cold fill and do not have an integral heater. In the U.K., most washers have a heater incorporated and can be plumbed so that they take both hot and cold, or just cold water from the supply. In some U.K. homes, you'll find that there is only a cold outlet for the washer, so on top of the electrical problems you'd then have to run a separate hot feed for the washer too.
None of that is very encouraging, I'm afraid, but I hope it shows why when it comes to large appliances such as these, you'll likely be far better off just selling and buying replacements over here. You can now obtain the larger "American-style" refrigerators, washers, and dryers over here if you really need the extra capacity and have the space.
