I know it's probably not worth the hazmat tradeoff for such a rare event, but the F-16 has an EPU powered by hydrazine that can spool up in about a second.
The 787 and nearly every other commercial aircraft with powered flight controls [1] (fly-by-wire or traditional) has emergency power available via RAT and/or APU, and any fly-by-wire aircraft has batteries to keep the flight control computers running through engine failure to power supply being restored by the RAT and/or APU. Due to its unusually high use of electrical systems, the 787 has particularly large lithium batteries for these cases. There is no need for an additional EPU because the emergency systems already work fine (and did their jobs as expected in this case). You just can't recover from loss of nearly all engine thrust at that phase of takeoff. [2]
1. The notable exceptions to having a RAT for emergency flight controls are the 737 and 747 variants prior to the 747-8. In the 747 case, the four engines would provide sufficient hydraulic power while windmilling in flight and thus no additional RAT would be necessary. The 737 has complete mechanical reversion for critical flight controls, and so can be flown without power of any kind. There is sufficient battery power to keep backup instruments running for beyond the maximum glide time from altitude - at which point the aircraft will have "landed" one way or another.
2. There is only one exception of a certified passenger aircraft with a system for separate emergency thrust. Mexicana briefly operated a special version of the early 727 which would be fitted with rocket assist boosters for use on particularly hot days to ensure that single-engine-out climb performance met certification criteria. Mexicana operated out of particularly "hot and high" airports like Mexico City, which significantly degrade aircraft performance. On the worst summer days, the performance degradation would have been severe enough that the maximum allowable passenger/baggage/fuel load would have been uneconomical without the margin provided by the emergency rockets. I'm not aware of them ever being used on a "real" flight emergency outside of the testing process, and I think any similar design today would face a much higher bar to reach certification.