When we bought our 23' Safari the previous owner told us she had burned out the 600 watt inverter. I said that most likely it just blew some fuses. Once home I opened it up and replaced 2 internal fuses. In the process of removing the inverter I shorted the positive wire to ground and blew the very expensive in-line fuse. I had to remove most of the L-couch to get to that so I could replace it.
My inverter has a plug that allows 110v to pass through when you're on shore power but there's no 110v plug near it, so we just use that inverter wall plug when we're not on shore power, nothing else. We have other 110v plugs near the inverter outlets and I just use short extension cords or power strips.
If your inverter light isn't coming on then something is not right with the inverter. I don't know how much access you have to the inverter but you need to get to it and check it out. See if the
12v leads from the battery are powered. If not, then the very expensive inline fuse is blown. (Search online for good prices.) If you have 12v+ and ground going into the inverter then the internal fuses are blown and need to be replaced. Mine were an odd amperage but I found replacements on Amazon.
If you don't have a multimeter so you can check voltages then I suggest you just disconnect the batteries, remove the inverter and take it to an electrician / handyperson to check it out.