I've been using a 36V motor that draws ~13A current at peak. I initally used a set of three lead-acid (12V 12A) batteries in series to power the motor. They had sufficient power to easily take a payload of more than 60kg and output ~35km/h at full speed. I have now replaced them with a set of Li-ion batteries (10 x 12.8Ah + 10x 12.8Ah = ~26Ah joined using short copper wire strips ~1cm) however, I am worse off than the lead-acid batteries. They can't even provide enough power. Without the load, battery voltage stays constant at ~4.2 but when I put load on it, battery instantly drops below 3.8 and is unable to provide neither speed nor power.
Can anybody tell what is going on and how can I get the power from the batteries?
The UltraFire batteries shown in your photo (and in your calculations) are fake. It is impossible to have 12,800 mAh capacity 26650 batteries. This is similar to my answer here about 18650 batteries. The largest capacity 26650 batteries currently available are less than 6000 mAh.
Not all Li-ion / LiPo batteries are equal in their ability to deliver all the energy inside. Currently, with 18650 / 26650 batteries, there are effectively "medium capacity, high discharge rate" battery models, and "high capacity, lower discharge rate" battery models.
If you get the highest capacity batteries, you are unlikely to be able to discharge them as quickly as some lower-capacity batteries. This is not my specialist area, but from the research I did a while ago, there appears to be a trade-off between the battery manufacturing processes for these two features - capacity and power delivery (effectively the internal resistance). This is a topic frequently discussed in flashlight and RC model forums.
You need to investigate the "C" discharge rating of whichever Li-ion / LiPo batteries you are considering, and read their datasheets very carefully, to understand how quickly you are allowed to discharge them (safely). Just looking at their capacity (in this case, the fake 12,800 mAh) does not tell you how quickly you can discharge them, nor does it tell you their internal resistance (which is related), even if the capacity value was true (which it isn't).
Building your own battery packs, without any form of BMS or other protection (as shown in your photo) is high-risk, due to the lack of balancing or voltage control for the individual cells. That is a whole topic on its own.
(P.S. Soldering directly to 18650 / 26650 batteries does them no good at all, and can easily damage the internal plastic separator sheets, leading to reduced capacity or even overheating, fire and venting. Can people get away with doing that sometimes? Yes. Is there a risk in doing that? Yes, you will see warnings on 18650 / 26650 battery datasheets not to do it. There are good reasons why professional battery pack manufacturers don't do that.)