Head, Shoulders, Knees and Toes, Knees and Toes

JavaScript (ES6), 91 88 87 bytes

n=>'knees,toes,head,shoulders,eyes,ears,mouth,nose'.split`,`[(245890>>(n%22&~1))&6|n%2]

How it works

We have 4 distinct pairs of words that always appear together: 'head' is always followed by 'shoulders', 'knees' is always followed by 'toes', etc.

Therefore, we can use the following index:

00: [ 'knees', 'toes' ]
01: [ 'head', 'shoulders' ]
10: [ 'eyes', 'ears' ]
11: [ 'mouth', 'nose' ]

And compress the whole sequence (in reverse order) into the following binary mask:

00 00 01 11 10 00 00 01 00 00 01

We use [ 'knees', 'toes' ] as the first pair to get as many leading zeros as possible.

We pad this sequence with an extra 0 so that the extracted value is premultiplied by 2, which leads to:

0b00000111100000010000010 = 245890

Hence the final formula for the correct word:

(245890 >> (n % 22 & ~1)) & 6 | n % 2

Test cases

let f =

n=>'knees,toes,head,shoulders,eyes,ears,mouth,nose'.split`,`[(245890>>(n%22&~1))&6|n%2]

console.log(f(0));    // head
console.log(f(1));    // shoulders
console.log(f(7));    // shoulders
console.log(f(13));   // ears
console.log(f(20));   // knees
console.log(f(35));   // ears
console.log(f(37));   // nose
console.log(f(98));   // knees
console.log(f(543));  // nose
console.log(f(1000)); // knees

05AB1E, 36 35 34 bytes

“‡ä¾ØsÏ©s¸±s“#2䤫Г—íÖÇ©¢ÄÓ#s)˜è

Try it online! or as a Test suite

Explanation

“‡ä¾ØsÏ©s¸±s“                        # dictionary string 'head shoulders knees toes'
             #                       # split on spaces
              2ä                     # split in 2 parts
                ¤                    # get the last part ['knees', 'toes']
                 «                   # concatenate and flatten
                                     # STACK: [['head', 'shoulders'], ['knees', 'toes'], 'knees', 'toes']
                  Ð                  # triplicate
                   “—íÖÇ©¢ÄÓ        # dictionary string 'eyes ears mouth nose'
                             #s      # split on spaces and swap top 2 elements of stack
                               )˜    # wrap stack in a list and flatten
                                 è   # index into list with input

In short, we build the list ['head', 'shoulders', 'knees', 'toes', 'knees', 'toes', 'head', 'shoulders', 'knees', 'toes', 'knees', 'toes', 'eyes', 'ears', 'mouth', 'nose', 'head', 'shoulders', 'knees', 'toes', 'knees', 'toes'] and index into it with input (0-indexed).

Python 2, 158 148 137 128 114 109 104 bytes

Lookup table seems better. Also shortened the big string and reordered the items. -5 bytes thanks to Rod for using string as a list.

c=int('602323'*2+'4517602323'[input()%22])
print"smkteehnhonoyaeooueeerasutesssdelhs"[c::8]+"ders"*(c<1)

initial solution:

n=input()%22
n-=10*(n>15)
if n>=12:n-=8
else:n%=6;n-=2*(n>3)
print"hskteemnehnoyaooaoeeerusduessste ls   h  d       e       r       s"[n::8].strip()