As Lilly lay down tucked up in her bed, all the days events span round in her head.
As Lilly lay down tucked up in her bed, all the day’s events span round in her head.
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Spun round is correct, as is all the day’s events.
The use of the apostrophe in day’s is correct, as you are referring to all of the events of the day. Likewise, saying all of today’s events is also correct.