I found this paper handy to get up to speed, at least up to 2005 technology:
https://www.electro-tech-online.com/custompdfs/2012/04/DanielW.pdf
JimB's path calculations are correct by my reckoning.
It might be obvious, but worth a mention, that the nature of this transmitter that must be built is entirely dependent on the receiver that is up in space. That receiver dictates the frequency to be used, the modulation type to be used, and the data encoding necessary to communicate. If the thing in space is an American government craft of some sort, I would think that it more likely that the control frequency would be somewhere in S band.
I quote from another paper (see ref 2) "existing deep space receivers must phase-lock to the uplink signal and generate a data clock for command reception. These processes limit the sensitivity of present-day receivers to about -157 dBm for carrier lock and -145 dBm for reception of the minimum bit rate of 7.8125 bits per second. " The example mentions an uplink frequency of 7.2 GHz, which I take to be used for the command radio. This is X band rather than S band. The reference also provides some interesting example data that might be considered realistic including:
For interplanetary contact:
ground station dish 10m
ground station tx power 1KW
ground station antenna gain 55 dBi
path loss at 7.2GHz 289 dB
atmospheric loss 0.2 dB
spacecraft antenna gain 7 dBi (typical low gain of a command antenna)
total received power at spacecraft -170 dBm
spacecraft receiver noise temperature 324K
received P/No 3.5 dBHz
In order to bash together a transmitter capable of transmitting in this scenario one would have to have access to some reasonably good instrumentation, something a serious hobbiest might have. Items might include a signal source such as an old HP 8672A signal generator (bare bones microwave generator well within the reach of serious hobbiest budget at around $1000) to which a BPSK or QPSK modulator (or something simpler. what kind of modulation is typical for spacecraft low rate command channels? I don't know) and a power amplifier could be added to form the transmitter. A modulator would be built on a circuit board (by a skilled RF engineer such as our heroine). A power amplifier is tricky, so I would use a bit license (and the fact that we are only communicating 2 million miles which isn't a lot in space) and target an output power of, oh, say about 100 watts. I think this is do-able at 7GHz, but a bit of a stretch. Its much easier to get 10 watts from a few good semiconductors in a carefully designed PA. I'm not sure how I would get a hundred watts. The antenna would be a dish of course, and probably a surplus older TVRO dish (or one of those neat ones that you see at the local cable company head-end compound) modified to put our power amp and a feed horn up at the feed point.
To get everything to work, a skilled RF engineer would need, at a bare minimum, a spectrum analyzer and probably a vector network analyzer to build the necessary hardware. Serious hobbiests have such things, usually older HP gear such as HP 8593E 22GHz spectrum analyzer and the venerable and much loved HP 8753ES vector network analyzer. These two can be a bit pricey at, say, $4000 for the SA, and $10K for the VNA, but a handy engineer would borrow such things for a single project. (EDIT: just remembered the 8753 series only goes up to 6 GHz. Better use an E5071C instead.)
ref 2:
https://www.electro-tech-online.com/custompdfs/2012/04/A0067Paper.pdf