There is no way that you can make your elbow heal faster. Nature does that for you hence the splint so that you cannot move it while it heals. As for sleeping, you will probably have to try to sleep on your back, but only you know what is comfortable for you. Did you know that an elbow fracture is one of the most painful fractures (that is a full fracture, not a hairline one) that you can get. Be patient, it will heal within a few weeks.

You can't make it heal fast; you need to accept the recommended treatment of splinting. You can take some medication for pain before bedtime and support your arm on a pillow. Be prepared to wake up a few times overnight because you will be either uncomfortable or the splint will inhibit your normal movements in your sleep. Keep your pain medication and some water close to your bedside in case you need it. Try not to worry. Your elbow will heal.