You spell it with one ‹t› because if it were spelt with two, it would rhyme with hitting instead of with fighting.
As for how “hard” your ‹t› is, compare these:
Notice how only the last one has a “real” ‹t› in the middle of it.
Edit: Integrating some comments.
You’re right that the ‹t› in writing is different from the one in written. The one in writing is still heard, although it is often a simple flap:
However, the ‹t› in written often reduces to a mere glottal stop:
[ʔ]. That means it is not heard; the glottis just stops moving for a moment.
/t/ in English has several different allophones, which vary by word and sometimes by speaker. Expect to hear
/t/ realized as any of
[ʔ], depending on various complex factors.