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Using a goniometer is primarily intended to measure the range of motion in a joint, specifically the degrees of flexion and extension. In the context of severe osteoarthritis, the condition often leads to stiffness and decreased mobility in the affected joints, making it crucial for the nurse to assess how much movement the knee joint can achieve.

Measuring the degrees of flexion and extension provides valuable information about the functional limitations the client may be experiencing due to osteoarthritis. This measurement can help in planning appropriate interventions, monitoring progress over time, and determining the need for physical therapy or surgical options.

The other options do not utilize a goniometer effectively:

- Evaluating joint stability involves assessing the structural integrity of the joint, which typically requires different clinical assessments or imaging rather than a goniometer.

- Assessing muscle strength is performed through manual muscle testing or other strength assessments, not with a goniometer.

- Joint swelling is generally assessed through observation or palpation rather than with a goniometer, which specifically measures angles of movement.

Thus, the accurate application of a goniometer in this scenario focuses on capturing the extent of movement in the knee joint, which aligns with the correct answer.