On the lower section of the roof, put a bigger kick on the first course, pack the baton with scraps of lead to gain height.
Grading your slates will also help.
Can't really say without knowing specifications, but if you don't have a rafter in the centre, you will have an aperture of 1.6m
i.e. 4 crippled rafters. Concrete tiles are heavy!
If they didn't put any sealant on the cill at both ends, then water can go sideways, and enter the property. Looks like some body has already had a go at this problem before looking at those bits glued on.