I Drive is a long road, but the main attraction is the shops and restaurants between Westwood Blvd and Universal Blvd. Around here, there aren't many hotels with washers/dryers in the rooms - I can't think of any that would offer that at the moment.
You'd need to look at suites or apartments for that - maybe somewhere like The Point, which is just off I-Drive near Universal, or further south on I-Drive, somewhere like Floridays, Sheraton Vistana or Hilton Grand Vacation Suites. By this time you're away from anything within walking distance, which - for us - is the whole point of staying on I Drive.
We have stayed on I Drive numerous times and used the laundry rooms onsite at various hotels. You don't have to sit with your laundry. We put the laundry in during a pool afternoon. The washers in hotel laundry rooms are usually huge and don't take much time. A wash and dry costs around $5 with detergent (or just bring some detergent tablets form home if you prefer). We have never had to wait for a machine in all our trips. It's not something we would base our choice of hotel on. We would base our choice of hotel on location, though, and for us I Drive is perfect.