The control board for these is typically based around a TT6061 touch/timer/control chip, and a BT134 or BT136 Triac to turn the lights on and off or dim at any one of 3 levels. I suspect you will have the lower rated BT134 device, as the bulbs are limited to 40 watts. What often breaks the Triac, is the high current “plasma arc” that happens inside a bulb when the tungsten filament fails. I have found over the years of repairing these, that the best bulbs to use are the Halogen types, as the little Halogen lamp cartridge (28w or 42 watt) that is inside the outer glass (decorative) envelope, has a gas mixture under a slight pressure, so when the filament fails, there is usually NO arc at all, to break the electronics. You CAN get the Triac, and replace it, but sometimes the IC chip get broken too, and they are difficult to get, and to replace, unless you are good with de-soldering and re-soldering techniques. It is often easier to replace the touch module, if you can get one. Check on EBAY or a good electronics shop.
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Once fixed, use LED bulb. Low wattage same light.
FYI: if the touch lamp is the dimming type, need to ensure you get DIMMABLE LED lamps. It sounds like the particular lamp might be an unusual design, that turns each of the 3 lamps on, to have 1, 2 or 3 on, then all off. If so, then NON-Dimmable ones MIGHT work OK.
von Tony Davis