The kazoo is a 'singing drum', allegedly invented by an African American named Alabama Vest around 1840 in Georgia and allegedly the only instrument invented in the U.S.A. Similar instruments have been in use in Africa for hundreds of years.
Some early jazz bands used metal kazoos to sound like cornets or trombones. National Kazoo Day is 28th January (seriously).
Don't blow! Just imagine the tune and hum or vocalise it into the large end. Players can produce different sounds by singing specific syllables such as doo, doo, doo, or brrr, brrr to the tune into the kazoo.
Why not use your spare hand to add rhythm with a shaker?
Stoneware and glass jugs were common rhythm instruments. But why not make your own from a 500 ml milk bottle?
Play like it is a brass instrument. Put your tongue behind your lips, purse your lips and make raspberry noises into the jug. Skillfully vary the shape of your tongue, mouth and amount of breath for each note and vary the angle of attack to give different timbre to the sound. Tight lips and small opening give a high pitch, loose lips the low bass (and cost lives).
Don't forget your spare hand to add more rhythm with a shaker.
Bones are a very ancient percussion instrument. Visit your local butcher to get some cow ribs - cut from the far end of the rib away from the backbone about a quarter-inch thick, 1 inch wide and 7 inches long.
Spoons make a good alternative. Curl your index finger and grip the top spoon very firmly between thumb and index finger across your knuckle.
Hold the bottom one, upside-down, firmly between index and middle finger a little further along the handle.
Let the bottom spoon move as you hit your hand, leg, arm or whatever takes your fancy.