Hi~

This is Teacher Rach~ Here's your correction:

I almost couldn't get up this morning.

I heard the clock chime but I slept a little late.

==> I heard the clock chime but I slept a little longer.

So I've had a very busy today because of starting late.

• If you mean you kept or continued sleeping, it's better to say you 'slept longer'. 'Slept late' means you went to bed later than usual.
• Good job for using new vocabulary appropriately! Keep posting~~