Even if he just got really bored and wanted hardware, I don't think he would have the knowledge to do it. It is one thing to know how to use all the tools, it is another thing altogether to know how to engineer high quality tools with nothing but sticks, stones, and no help.
Of course he would stop and build a computer first. And universities where people would read his books, banking and mail institutions to send people checks through after they report bugs, and so on. One needs to have some preconditions in place first, obviously.
As I see it, the hardest part is making the magnets to generate electricity from a water mill. Then you only need some copper and to make a lot of wire out of it, and you can make electro magnets for the gates. (It seems to me to be even easier to build than Babbage's mechanical computers).