I think it's awesome that you are making music with your friends and playing with a band. I can understand what you are trying to accomplish. But I don't believe you will be happy with the results from a 15 watt amp speaker combo. Trying to provide sound reinforcement for vocals, harmonica, drums and guitar with a 15 watt amp is simply asking way too much out of the amplifier for good results.
You should be able to get all your mics and guitar to emit sound through the amp's speakers, but the quality and volume of the overall sound is not going to be stellar by any stretch of the imagination.
This setup might work fine in your garage or living room, but in that scenario, you won't really even need amplification. And as Tim mentioned, miking drums to play through a 15 watt amp makes no sense. I don't even mic the acoustic drums in my band playing through a 1200 watt PA system unless we are playing an outdoor gig where the demands are greater due to no walls to reflect the sound back and greater ambient noise (traffic, airplanes, wind blowing, etc.). In most places we play (restaurants, sports bars) the drummer has to hold back or use hot rods or brushes to keep from being too loud in the mix.
Most 15 watt amps are optimized to do one thing well - amplify a single guitar which in the absence of an amplifier, would not be heard. The speakers are not intended to be "full range" as it is anticipated by the manufacturer that the only frequency that needs to be reproduced, is that of an electric guitar.
There are some two channel acoustic amps which are designed to amplify both vocals and a guitar, but the smaller less powerful versions of those yield the most satisfactory results with just one mic and one guitar.
You will get more useful results from a small powered PA speaker with at least 100 watts. In general, the more powerful PA speakers will sound better. You could probably find a used Mackie Thump 1000 watt 12 inch full range powered PA speaker for about $175 - $200 US or a Behringer Eurolive B110D 200 Watt Powered Speaker with a 10 inch speaker used for around $150. US. All of these Powered Speakers have inputs for both a guitar and microphone and you can feed a mix into them through a small unpowered mixer as you are contemplating.
This speaker by Harbinger would also work and it is often on sale new for $150. (I have one) Harbinger Powered PA Speaker at Guitar Center
And if you shop around - you might find a deal like this for the Harbinger Powered PA Speaker