First, I think it would probably be easier to send an alert to the admin when the service stops, and have the LEM try to restart it immediately.
Second, the LEM works by responding to events and logs. There are no events or logs for "something didn't happen," since that would lead to some weird logs:
1424788635000 188.8.131.52 ASA-8-2715685 The Universe did not convert to something made entirely of lasagna in this second.
1424788635000 184.108.40.206 ASA-8-2715686 You did not turn into a giant lizard in this second.
1424788635000 220.127.116.11 ASA-8-2715687 The word for 'Tuesday' was not removed from your language in this second.
1424788635000 18.104.22.168 ASA-8-2715688 A hamster did not appear in Sandra's lunch in this second.
However, the LEM can try and look for the absence of events. Maybe something like this? This rule required you to define a Time of Day Set for your 10:00 AM window.
Edit: I went and talked to one of my engineers, and while the LEM can handle certain "NOT" events, it needs something to occur to "start the clock" for watching for the NOT event, so this rule probably won't work. It'd be best to just have the LEM respond to the initial stop message immediately.
You can create this rule exactly as described -
ServiceStart NOT EXISTS
ServiceStop.ServiceName = ServiceStart.ServiceName
Response Window: 60 minutes
If you want to refine it only to a certain service, add another condition:
ServiceStop.ServiceName = <service to watch for>