The KX1 is a nice little rig. I don't use mine very often, usually preferring the KX3 over it, but from time to time it's fun to operate with a relatively minimalistic rig that you can assemble yourself (including winding inductors and soldering all the components). Of course its old-school technology (NE602 mixer etc.) can't be compared to SDRs with QSD/QSE like the KX2 and KX3, and power output is lower at 3-4 Watts as well, but its simplicity is intriguing, and on those SOTA activations where every gram/ounce counts, its lower weight is certainly a plus. Also, I like the fact that it can be equipped with a built-in ATU, something missing from the otherwise quite similar MTR rigs.
Built into the KX1's case are two 3xAA battery holders, intended to hold six Alkaline or NiMH cells. Unfortunately, power output from the KX1 with voltages around 9 V (or even just 7.2 V in case of NiMH) is quite low, only 1-1.5 Watts. Also, if you use rechargable cells, you need to take them out of the case to recharge.
I've been looking for a better solution that would provide more voltage with as little weight as possible, and allow for charging without opening up the rig – for example when abroad in a hotel.
Nowadays, the battery chemistry of choice for portable ham radio operations seems to be LiFePO4. These are safer than traditional Li-Ion batteries and have a more suitable nominal voltage. 4 LiFePO4 cells in series yield a nominal 12.8 V, or around 11.2 V when discharged. Aside from assembled battery packs (popular with portable ham operators), these cells are also available individually in AA size on eBay. Various brands are available; the plain silver ones labelled with "MPIFR" seem to be of reasonable quality (e.g. this auction here, there are other sellers). Beware of AA LiFePO4 cells claiming more than about 700 mAh capacity; this is most likely fake (the reasoning behind the inflated figures seems to be to give an equivalent charge to 1.5 V batteries).
Lithium cells need a protection circuit to protect them from overcharge, overdischarge and overcurrent, and to keep the individual cells in a pack balanced, as they can fail quickly otherwise, or even burn or explode. Suitable protection circuit modules can also be found on eBay; the ones labelled DLG-3004 fit nicely into the KX1 case and appear to work fine in this application (e.g. this auction here, there are other sellers, search for "protection circuit LiFePO4").
I've found the quiescent current of the protection module to be higher than indicated in the eBay auction details, with a measured 26.5 μA. But that's still low enough that the batteries are safe if recharged about once a year.
The KX1 case is quite cramped, but if placed in the upper left corner of the bottom part of the enclosure, the aforementioned protection module fits into the case even when the ATU option is installed. Use (thick) double-sided adhesive tape to install the module, which nicely isolates the bottom of the module from the case. The battery connector that goes to the KX1 main board needs to be cut and connected directly to the P+/P- terminals on the protection module.
The protection module needs connections to each individual cell in order to monitor/balance the cells. If the batteries are installed as shown in the picture below, two of the connections between the cells can be accessed by soldering a thin wire to the black wire that runs along the bottom of each battery holder (scrape off some of the insulation, solder and then isolate the joint with Kapton tape etc.). The remaining connection can be made at the splice that connects the two battery holders. In this way there is no need to solder to the spring wires of the battery holders, which can be tricky. The two unused battery holder positions are filled with dummy cells (also available on eBay, often supplied along with the AA LiFePO4 cells).
Take care to connect the protection module correctly. B- connects to the negative side of the last cell in series (the black wire from the right-hand battery holder that used to go to the KX1 main board), and B+ to the positive side of the first cell (red wire from the left-hand battery holder that used to go to the main board). B1 then connects to the black wire at the bottom of the left-hand battery holder, B2 to the splice between the battery holders, and B3 to the black wire at the bottom of the right-hand battery holder.
The KX1 comes with two diodes (D2 and D3) that prevent current from flowing between the DC jack and the internal battery. In order to charge the battery without removing it, diode D3 needs to be short circuited as shown in the picture below. In this way, the battery can be charged by applying a current-limited source of no more than 15 V to the DC jack, and there will still be no current flow from the battery back to the DC jack.
For a suitable portable charger, look for small "wall-wart" style lead-acid chargers. These have a suitable output voltage (around 14.7 V, which is perfect considering that there is still the forward drop of one 1N5817 diode in the KX1's circuit) with current limiting. Make sure to pick one with a current limit of around 600-700 mA so the cells do not get charged too quickly. For "smart" chargers that check the battery voltage before applying power, you will have to short out D2 as well (which will of course make the DC jack unsuitable for use with external power sources except for a charger, and also get rid of the diode forward voltage drop). The protection circuit module used here does not include a voltage/current regulator, so a simple external power supply will not work, and will either charge the batteries with a much too high current (and/or blow D2 on the KX1 board), or get overloaded when the batteries are empty.