Your wife would claim the expenses as her business expenses. She is considered to be self-employed, and would file as such. When going through TurboTax self-employment section (so at this point she's considered to own her own business) you'll be asked to enter self-employment income with or without the form 1099-MISC; it is called General Income/Business Income. Simply enter her cash/check income in that section. You're correct -you can't just fill out a W-2 for the job, nor can you fill out 1099-Misc since she didn't receive it.
Then, when going through the Business expenses, you may enter your wife's expenses for this job. If the expenses descriptions do not match what she has, you will be asked to enter description and amount on your own, in Other Common Expenses section.
