It is important to appreciate that both the motherboard and the power supplies suffer from this problem. If the computer will not start, but the front panel LED flashes amber, the power supply has failed. Many people just recommend replacing it, but these power supplies are not industry standard - you may have to modify the case with an industry standard replacement supply.
The good news is that in most cases the power supply is repairable - it's just the low voltage high value capacitors again! Here is the process I followed.
1. Disconnect from supply, open case and remove power supply from tower
2. Open up power supply. You will see the offending capacitors around the point where the cables exit the box.
3. Several of mine were either leaking or bulging, or both. I carefully noted the polarity for each and replaced the following:
3300 Microfarad 10V 105deg C 12mm diameter (RS 526-1171) - one of these
4700 Microfarad 10V 105deg C 12mm diameter (RS 526-1222) - one of these
3300 Microfarad 16V 105deg C 12mm diameter (RS 526-1345) - two of these
2200 Microfarad 10V 105deg C 10mm diameter (RS 527-417) - three of these
1000 Microfarad 16V 105deg C 10mm diameter (RS 571-656) - two of these (one was originally 10v so 16v does fine)
You need a small soldering iron, some desoldering braid and a little patience
all these available next day delivery from RS Components
Power supply now working fine again, and no tower modification was needed. A good day, a good job with a system and its licences saved from the scrap heap.
All this presumes that the safety fuse is still intact (check with a meter). If it is not, the high voltage side (control transistor) has blown and then you will have no choice but to replace. But this is rare.