The problem is that toaster, Keurig, microwave are all high power devices. They may only need power for a few minutes at a time, but when they do need power, it is in the several hundred watt range. Battery power (short of Tesla PowerWall) won't do you much good.
This is actually why code calls for a minimum of 2 countertop circuits in a kitchen. But many older kitchens (including mine before renovation) simply don't have that - they often have just 1 circuit, sometimes shared with other rooms.
You have learned why extension cords are not a good idea for long-term usage. The only really good fix is to get additional circuits run to the kitchen. Since this is a rental, in most places this needs to be done professionally. Not DIY, not by your landlord, but by you (with landlord's permission) or your landlord hiring a licensed electrician.
Contact your landlord. If they won't help - perhaps claiming (often correctly, but not always) that the kitchen is OK because it is grandfathered under older code. The catch is that pushing harder on the code issue (e.g., almost certainly applies if the kitchen was built or renovated in the last 30 years) may result in your landlord finding an excuse (often not hard to do) to kick you out, or to drag things on for a while until renewal time and then simply not renew your lease. It gets even trickier if it this is an unofficial rental. So things can get tricky and beyond a simple request you may not want to pursue it.
The next best thing is to use an extension cord that never goes down to the floor and that is of the correct size - minimum 14 AWG for a 15A circuit, minimum 12 AWG for a 20A circuit. I would use a short cord (probably 3 - 8 feet long). Finding such a cord may be non-trivial - in a quick search, the short 12 AWG cords all have 5-20 connectors - i.e., designed for 20A receptacles and devices. But you could have 15A receptacles (legally) on a 20A circuit, and most of your devices will be 15A devices. So it gets complicated to do this in a reasonably safe way.
The temptation is to use a multiple-outlet strip like commonly used for computers. They are inexpensive and easy to find. The problem is that these outlet strips, even when reasonable quality, are designed for relatively low power usage. In fact, using them for microwave ovens and toasters can actually damage them because they are often designed for much smaller loads.
