My father was napping on the sofa with his feet stretched toward the fire.

Literal

Father [topic-は] both-feet [object-を] fire direction [to-に] stretched, sofa [at-で] napping was.

居眠りする means 'to doze off' — an unintended nap. 両足を火のほうにのばして is a vivid physical description using the te-form as adverbial manner.