The question is ambiguous in prime characteristic, since the meaning of
"semisimple element" isn't straightforward for a simple Lie algebra consisting of matrices. Possibly what's meant here is "semisimple as a matrix in the given realization of L". but there's no reason for this property to be intrinsic to the Lie algebra itself as it is in characteristic 0 (thanks indirectly to Weyl's theorem on complete reducibility). Maybe "semisimple" refers instead to the adjoint action? Anyway, it's ambiguous. Jordan decomposition works best when it propagates automatically to other matrix representations of the given Lie algebra.
On the other hand, Engel's theorem holds in arbitrary characteristic: if a Lie algebra consists of nilpotent (or ad-nilpotent) matrices, it must be nilpotent and therefore not "simple" when that's defined to exclude abelian Lie algebras.
But in general there's no reason to expect a simple Lie algebra of matrices to contain the semisimple and nilpotent parts of its elements. Moreover, even for restricted Lie algebras there is not yet a complete classification of the simple ones in characteristic 2 or 3. So the question asked needs to be more
carefully focused to have a yes/no answer.
ADDED: Given the clarification and comment, maybe it would be helpful to comment briefly on relevant literature (which is technical but not too plentiful in this area). For the structure theory of modular Lie algebras, with emphasis on the restricted case, there is a monograph by G.B. Seligman (Springer, 1967), especially V.7, and related papers by D.J. Winter such as the one in Acta Math. (1969). Then there is a harder-to-find Dekker book by H. Strade and R. Farnsteiner (1988), especially Chapter 2; this book was at first typeset readably, according to one of the authors, but Dekker insisted on replacing it with a typed photocopy to fit their style. Later work on classification of simple Lie algebras (especially restricted ones) appears in papers by Block-Wilson and later Premet-Strade, who also wrote monographs. As elsewhere, this type of classification tries to exploit classical ideas involving "toral" subalgebras.
At any rate, in restricted Lie algebras which are simple and live over suitable fields, one eventually knows that Jordan decomposition makes sense and propagates under restricted homomorphisms, while nonzero toral subalgebras as well as nonzero nilpotent elements exist. The focus on these Lie algebras is due of course to the fact that the Lie algebra of linear algebraic group has a natural restricted structure when viewed as an algebra of derivations.