I was induced at +11, DD was born +12. In all honesty, I look back and I would have made a different decision and done it earlier. I think at this point its worth thinking through your various options for how you would want an induction to go. I also wonder if they'll tell you your Bishop score? Because if its v low in your position I really would move an induction date sooner, whereas if its possible you'll go into spontaneous labour in the next day or so I would wait and see.
In my case, I wanted to go as long as possible to increase likelihood of a 'normal' labour and delivery. But my bishop score on the day of induction showed I basically wasn't at all receptive to labour. I was then overstimulated by the pessary, they had to take it out, DD was very distressed but it resolved. However it was a pretty awful labour: I experienced it as one long contraction, no breaks. Then, when I finally got to 3cm, they were again concerned about her heart rate: I had an epidural, waters broken (Meconium in waters)and the drip. Her heart rate went funny again, as I reached 10cm, and had instrumental delivery to get her out fast.
About 2 weeks postpartum, she developed a mystery illness and spent a week in hospital. I had group b strep and I will always wonder if the extra time in there caused the infection to travel to her. I look back and I think an earlier delivery would have been better.
Honestly, its anecdotal but I think I'd want to know whether you really are 13 days over or whether there is some uncertainty about dates because in my experience, some babies just don't seem to want to come out. Most people I know who went late then had a spontaneous labour and delivery were unclear about their dates, so probably actually went into labour a few days after their actual due date.
Personally if I did it again, I would go for induction earlier. At the moment, I have a planned section lined up, but that's really to do with the bad tear and the fact I've confirmed group b strep and am paranoid about the exposure of a vaginal birth. With a bishop score that low, I'd want a plan in place for ensuring I could move to the drip/epidural more quickly: they wanted me to wait the 24 hours/till I was 3 cm and I was exhausted as a result, and I suspect she was too.
I don't at all mean this as a horror story, my birth worked out absolutely fine, but I think I got too caught up in the 'your body knows what to do' argument. The simple fact is, there used to be a much higher rate of stillbirth, maternal loss, and long lasting damage done to children. We absolutely have too much unnecessary intervention, but its really hard to know where to draw the line. In my case, looking back, I think I'm not happy with the risks I took. I got too caught up in 'the hospital will push intervention, its your job to resist it.' Its really difficult to make a decision when it feels people aren't weighing up your individual risks but are taking a 'computer says no' attitude. Personally, I don't know anyone who has gone past 14 days. I think that would be my cut off in terms of 'rewards' vs risk - I really think a low intervention spontaneous labour isn't very likely at that point, and the worst case scenario risks are quite scary
I'd also try to focus on what you want to feel like when the baby gets here. I was pretty exhausted after my labour, it definitely had an effect. This waiting will be really tiring you too. I know it can feel like pregnancy is all about leading to the birth, but this bit isn't really the fight: its the weeks ahead and adapting to the baby and you want to be in the best shape for that too.