The grid between the palace and the Plaza de la Constitución holds nearly everything — small hotels in 18th-century courtier houses. The Mercado de Abastos corner is best for restaurants. Nothing is more than 15 minutes' walk from anything, palace included.
Aranjuez lists around 24 bookable hotels and guesthouses, from roughly €52/night. Prices on the area page are live; booking 3–6 weeks out usually lands the best rate, with free cancellation on most rooms.
If it books out or you want to compare: Chinchón (from €55/night), Lavapiés (from €35/night), Alcalá de Henares (from €48/night) — each with its own area guide on this site.
Curated picks are coming — meanwhile, the live search covers every bookable property at the same price or better.