Here you go. If you can't suspend disbelief there's a simple explanation:
It's very experimental, and hasn't been tested thoroughly. It's sectioned so that it'll wrap around the body and can be stowed, and can even be considered extra armor if unused. It obviously requires a lot of power to stiffen from chain to sword, and it's probably one of the reasons there are two nuclear reactors onboard, rather than one.
All the power has to go through contacts of some sort on the hand to the sword hilt, and after all that fighting there must be some doubt that it would work at all, not only because of the possible contact damage, but also due to possible damage to the sword. This was an unusually large kaiju, and their survival wasn't due to being in a cool machine, but because they knew how to fight.
So it's not brought out in the beginning because it's untested, and it is brought out when there really are no other options because there are no other options.
Further, Mako is essentially riding for the first time - she isn't going to be thinking about that one experimental thing she added to the jaeger months ago, she's thinking about how she doesn't want to die, and how to make that happen, or how awesome it is to be driving and drifting for the first time. Drifting puts people in each others heads, and anything that floats to the subconscious level is seen by the other, but it isn't a mind reading machine - you don't get access to everything the person knows. You do get all their active feelings and thoughts.
If she didn't think about the sword until the critical moment, then neither did what's his name.
The only real plot hole in this whole sword issue is that you can bet what's his name would have read the docket concerning the upgrades to his machine. There's no way he wouldn't become intimately familiar with all the jaeger's new nooks and crannies. He obviously knew how to vent coolant - which may be an old feature, but he couldn't have taken that for granted with the upgrades. (speaking of which, if they were drifting he needn't have told her how to do it, or even to do it - that was just for story exposition - so if the sword thing bothers you, this should really bother you)
Maybe he just accepted "refurbished and with two nuclear power plants rather than one" and didn't bother to look into it further, or maybe he didn't want to think about it due to the issue with his brother in connection with that machine, so you might be able to explain it away, but I don't see that happening.
And if that isn't enough, the final explanation is, "Because it's in the script that way."
Now, using the jaeger action figure, please show the court where the bad movie touched you...