[Question] What would an AI need to bootstrap recursively self improving robots?

Picture the scenario: A malicious superintelligent AI creates a weapon that wiped out all humans instantaneously. The AI has complete control over all network connected devices and computers.

Most manufacturing will stop working immediately without humans to complete that steps that can’t be automated. Within minutes the first power stations will stop working. Even if the AI can get them to work eventually they’d run out of fuel, which can’t be extracted, transported and used to refill the power stations without a lot of manual labour.

Solar panels/​wind turbines should last longer, but even then at some point in the grid you generally need humans to carry out repairs. Repairs that are tricky without the dexterity and versatility that our fingers provide.

The AI is then on a clock to get the stage where it can use the existing infrastructure we left behind to create robots/​nanobots that are able to bootstrap more powerful robots/​nanobots etc. till it can achieve the level of sophistication required for it’s aims.

How could an AI go about that? Is this a hard enough step that it’s worth putting safeguards in place to keep it hard? What would make it easier for the AI?

Assume that software won’t be any issue—the AI can easily hack that. The question is what it can do with the hardware we leave behind.