> How do we determine the bias from a given {cathode} resistor value? ... I feel like it must be a simple matter of rearranging a formula or something
It isn't. Or at least, it requires data not normally available.
On the chart, assume a current, compute the voltage, and put a dot at that current and grid bias. Do this over a wide range of possible currents. You end up with a curved line (which is why it isn't a trivial calculation).
Now spot your B+ and draw your plate resistor line from there. The intersection of the plate resistor and cathode resistor lines is your operating point.
But for any 12AX7 bias point useful for driving any significant load (even a 470K hi-fi line input) the plate resistance will be so close to 62K that you don't need to sweat-out the real value. It could be as low as 33K except out there on the zero-bias line it can't swing. It can be over 100K, but at bias points with such low current that it can't do useful work. Inside a high-gain amp where it feeds an inch of wire and a grid, it might work around 150K Rp, but not on a jack to the outside world.
So the output impedance is around 60K||100K||1Meg or 36K.
But that is not the load it can drive. As a general rule, a triode should see a load 2X to 5X its plate resistance. Lower makes distortion. Higher does not suck enough work out of the tube. So we want 120K to 300K load. We already blew that: the 100K load. Not much too low, and we may not ask a guitar amp to be super clean. But we don't want much added load. Over 10X the existing impedance, or 360K, would be nice (and is probably the design load). We could fudge that to 3X or 100K, with increased distortion and lower output voltage. However, in many cases a 12AX7 on simple audio (guitar, not guitar/bass/drums) is tolerable at lower impedances, and once you get down to Rp the distortion does not get worse, you just lose output voltage.
The transformer primary must be designed for the source impedance (36K) for bass response and iron distortion, for 100K-300K for low tube distortion in the bass. Taking 80Hz as the bottom of the guitar, this gives primarly impedance of 75H to 600H. You are not going to find a 600H winding. There has to be some compromise with bass distortion. That may not be bad in guitar duty, but has to be considered.
Do you really want to drive 600Ω? As Paul mentions, few inputs today are 600Ω. You can pretend you will only see 10K loads and usually be right.
If so: you need 8:1 with 75H to 22:1 with 300H. 10:1 is easily found, but either it won't have ~100H inductance, it won't pass line-level cleanly, or it will be exotic and expensive.
Taking 10K loads, you can use 2:1 at 75H to 5:1 at 300H.
You don't just want to match impedances; in fact impedance really should be a secondary concern. In post-1935 design, you put in enough tubes to give excess gain, so you don't HAVE to carefully impedance-match. How much level does this have, and how much do you need?
My guess is that this has enough voltage gain from guitar to make around 1V output in use, maybe over 3V peaks. So say 2V no-load, or 1V loaded in 36K. 1V in 36K is 0.027 milliWatts or -15dBm. You can NOT feed +4dBm 600Ω lines with this: not enough power gain from guitar input. Even assuming that loads WILL be 10K, output level is only -3dBu. That's within the gain-range of most "+4dBu" gear. So a good quality (high inductance) 40K:10K transformer will work. Line impedance is 5K plus the considerable leakage inductance of a hi-Z winding, so don't run it long distances. Even 20 feet might be "far".
If your goal is a mike input: any general-use mike input can take -50dB levels, say 2mV. You have 1V. 500:1 ratio seems called for, and would give extremely low output Z (0.15Ω!). We still need that 75H-300H primary, which is a tough winding. So take a common mike input transformer, backward. To isolate the back-to-back transformers, put a 6dB pad between. We need 4mV out, so with a 1:10 (used 10:1) we need 40mV in. And such a transformer likes to see 10K-20K source on the hi-Z side. We have 1V at 36K. We need a pad with >36K in, 10K out, and 1V/40mV loss. 250K and 10K resistors give the 25:1 loss, a 10K output Z, and a nice 250K input Z. The transformer can be plucked from any old-school gym PA system. Output level will be similar to a close-talked SM57, a level most board-ops understand.
If you must drive "line" inputs: try to find a good 40K:10K line transformer. They are rare beasties. And it won't be a great line output.
For an all-tube good line output: as Dave and Paul hint, you want another tube. 6C4 (half a 12AU7) is a safe choice, can work with inexpensive 10K:600 iron, give voltage gain of about 3, good solid +18dBm peaks.
Alternatively: add a 24VCT transformer and a 5532 chip. It kinda wants a TL071 also, because the input bias current of a 5532 makes 200K inputs difficult. Probably one of the hot FET-input chips with low THD and true 600Ω drive is simpler.