You suspect that the trigeminal nerve (CN V) is compromised and would expect there to be sensory deficits in the face.

To approach this problem, we need to think about a few things:

1. If the blink reflex is absent, what cranial nerves are involved?

The afferent part of the blink reflex is supplied by the trigeminal nerve (V), while the efferent action is from the facial nerve (VII).

2. The patient is able to make different facial expressions and shut her eyes tightly. Which cranial nerve is involved in facial expressions?

The facial nerve innervates muscles of facial expressions. The patient's ability perform these actions tells us the facial nerve is intact; thus, the trigeminal nerve must be the part of the blink reflex that is deficient.

3. What other function does the trigeminal nerve have?

The trigeminal nerve provides sensation to the face and scalp, and also innervates the muscles of mastication.

Absent gag reflex can be due to deficits in the glossopharyngeal nerve (IX, afferent portion) or vagus nerve (X, efferent portion). Lack of eye movement can be due to occulomotor (III), trochlear (IV), and/or abducens (VI) cranial nerves. Inability to turn the head is a result of sternocleidomastoid deficiency, which is innervated by the accessory nerve (XI). An impaired sense of smell suggests olfactory nerve (I) deficit.

The biceps brachii is innervated by C5 and C6 nerve roots. So, in the presence of a C5 nerve root irritation, biceps tendon reflexes may be diminished. the triceps brachii is innervated by C7 and C8 nerve roots, and the lower extremity muscles are innervated by L2 through S2 nerve roots, so testing these muscles would not tell us anything about a C5 nerve irritation.

The Romberg test involves maintaining an upright stance with feet together, which tests the vestibular portion of the vestibulocochlear nerve. The accessory nerve controls shoulder girdle and neck movement, the olfactory nerve provides the sense of smell, the hypoglossal nerve controls tongue movement, the facial nerve controls the muscles of facial expression (amon other functions) and the trigeminal nerve controls the muscles of mastication (among other functions). Thus, the Romberg test would not test any of these functions

The triceps brachii deep tendon reflex tests the C7 and C8 nerve roots, so testing it would not provide any information about a lumbosacral nerve root pathology. The cremasteric reflex tests the T12 and L1 nerve roots, the Achilles deep tendon reflex tests the S1 and S2 nerve roots, the anal wink reflex tests the S2, S3, and S4 nerve roots, and the patellar deep tendon reflex tests the L3 and L4 nerve roots.

Cranial nerve (CN) III (oculomotor nerve) is responsible for certain eye movements. It innervates the following muscles: superior rectus, inferior rectus, medial rectus, and inferior oblique. It also functions in pupillary constriction, accommodation, and eyelid opening (via levator palpebrae). CN III is a motor nerve along with parasympathetic function. If there is injury to CN III these functions will be impaired leading to the presentation of the observations in the question stem.  

The ulnar nerves serves the dorsal and palmar surfaces of the hand from the midline of the ring finger over to the little finger (dorsal and palmar).

The axillary nerve supplies three muscles, the deltoid, teres minor, and the long head of the triceps. The deltoid is a muscle of the shoulder and teres minor is a muscle involved in the rotator cuff. The axillary nerve also supplies sensory information from the shoulder joint, and to the skin below the deltoid. The axillary nerve may be injured during an anterior dislocation of the shoulder joint or a fracture of the surgical neck of the humerus. 

The radial nerve supplies the posterior portion of the upper limb. The radial nerve innervates the medial and lateral head of the triceps brachii muscle (the long head is supplied by the axillary nerve) and anconeus. The radial nerve originates from the brachial plexus and consists of fibers of nerves C5, C6, C7, C8, and T1. 

The ankle jerk (Achilles) reflex tests the nerve root S1, S2. Biceps reflex tests roots C5, C6. Brachioradialis reflex tests C5, C6, C7. Triceps reflex tests C6, C7, C8 nerve roots. Patellar (knee jerk) reflex tests the L2, L3, L4 nerve roots.

The pupillary light reflex and ocular accommodation reflex involves cranial nerve II for the sensory aspect and cranial nerve III for the motor application. The jaw jerk reflex uses cranial nerve V for both sensory and motor. The corneal (blink reflex) utilizes cranial nerve V for sensory and VII for motor. The gag reflex utilizes cranial nerve IX for sensory and X for motor.