That's not a lot of power coming from it to operate those 3 fluorescent lights so it probably will only get warm to the touch, not hot. A lot of devices now that are totally enclosed in plastic housing just have the 2 prong plugs, no 3rd prong for the safety ground. I guess the reasoning is that since the device is totally enclosed in plastic you can't get shocked from touching it. But that doesn't make your metal slot machine chassis grounded for safety.
The machine was made for the 3rd prong safety ground wire because the machine is made of metal and if a short were to happen inside it you could get shocked. The only way to fix that if you want to is to somehow run a wire (green is the usual color) from the machine metal chassis over to your electric wall outlet. You could get a power cable, like a short 16 or 18 gauge extension power cable that has 3 wires (black, white and green), cut off the female end and breakout the 3 wires. Connect the black & white wires (the power wires) to the power converter's power input, then connect the green wire from the extension cable to the machine chassis, like with a lug screwed into the metal. This is what they do on a clothes dryer power cable, and other appliances as well. There may already be some green wires connected to chassis ground inside the machine near where the power cable originally came into the machine, you could wirenut/splice the new green wire to one of those existing green chassis ground wires.
How are the fluorescent lights doing, any flicker or noise from them or their ballasts? Are the ballasts not getting overly warm?