You do not need another piece of hardware. If the Kasa app can turn it on and off, then theoretically Alexa should work with it.
I presume you have linked Alexa with your Kasa account and then asked Alexa to discover devices ? Also, you have to make sure that the socket to which the bulb is attached is turned on when doing this.
Without these steps Alexa will be blind to the Kasa smart bulb. If the bulb is still not discovered
you will have to reboot it and try again. I would suggest the entire process once again after disabling and then re-enabling the Kasa skill.