Learn Portuguese (Portugal): Eu tenho insónia, daí que durma mal.

The phrase daí que (meaning 'hence' or 'that is why') is a consecutive conjunction that always triggers the subjunctive mood. It links an established cause (tenho insónia) to its consequence.

Could I use por isso to mean the same thing?

Yes, por isso (therefore or for that reason) is a great alternative, but it changes the grammar. Unlike daí que, por isso is followed by the indicative mood. You would say ...por isso, durmo mal.

Why is the verb spelled durma with a 'u' rather than dorma?

The present subjunctive of dormir is formed from its first-person singular present indicative form, which is eu durmo. You drop the 'o' and add the regular subjunctive endings, giving you durma.

I often hear Portuguese people say they have insónias (plural). Is insónia (singular) also correct?

Yes, both are correct. While tenho insónia is perfectly fine and refers to the medical condition as a whole, it is extremely common in Portugal to use the plural tenho insónias when talking about experiencing sleepless nights.