It would be:
Laura no ha empezado a hablar todavía.
Todavia tells you need to use the Preterito perfecto tense. Once you know which tense to use, you have to conjugate according to your subject ( Laura).
the Preterito Perfecto is built this way:
Haber conjugated ( He, has, ha, hemos, habéis, han) + participio pasado ( in ADO for verbs in AR, in IDO for verbs in ER and IR), unless they are irregular.
Empezar is regular, so it will be Empezado.