Die provisorische Lösung funktioniert nur kurz, danach nehmen wir den festen Plan.

Because Lösung is feminine and in the nominative singular, and it’s preceded by the definite article die. After a definite article in feminine nominative, the adjective ending is -e: die provisorische Lösung. Compare:

• Feminine nominative: die provisorische Lösung
• Masculine nominative: der feste Plan
• Masculine accusative: den festen Plan → adjective ending becomes -en: den festen Plan

• funktionieren is the standard verb for “to function/work (as intended)” for things, ideas, solutions: Die Lösung funktioniert.
• arbeiten means “to work (as a person)” or “operate” for machines, not for abstract solutions: Die Maschine arbeitet, but not Die Lösung arbeitet.
• klappen is colloquial for “to work out/succeed”: Es klappt.
• gehen colloquially can mean “to work”: Das geht, but it’s informal and context-dependent.

So funktioniert is the precise, neutral choice here.

It means “only for a short time/for a short while.” It speaks about duration, not physical length. Synonyms/paraphrases:

• nur kurzzeitig
• nur für kurze Zeit
• nicht lange Slightly different nuance: nur kurz is everyday and concise; kurzzeitig is a bit more formal.

German requires a comma between two independent main clauses when there’s no coordinating conjunction. danach is an adverb, not a conjunction, so the two main clauses are simply separated by a comma:

• Clause 1: Die provisorische Lösung funktioniert nur kurz,
• Clause 2: danach nehmen wir den festen Plan. You could also write a period or semicolon instead. If you use und danach, then normally you don’t put a comma before und.

No. danach is an adverb meaning “after that.” nachdem is a subordinating conjunction “after,” which introduces a subordinate clause with verb-final word order and usually perfect/pluperfect for past reference:

• Danach nehmen wir den festen Plan.
• Nachdem die provisorische Lösung nur kurz funktioniert hat, nehmen wir den festen Plan.

German main clauses are verb-second (V2). When an element like danach is put in first position, the finite verb must come second, and the subject follows:

• Correct: Danach nehmen wir den festen Plan.
• Incorrect: Danach wir nehmen …

It can be idiomatic in the sense of “to go with/choose the plan,” similar to choosing from options (like Ich nehme den Salat). If you mean “adopt/follow/implement” a plan, common alternatives are:

• danach folgen wir dem festen Plan (follow the plan)
• danach halten wir uns an den festen Plan (stick to the plan)
• danach setzen wir den festen Plan um (implement the plan)
• danach gehen wir zum festen Plan über or greifen auf den festen Plan zurück (switch/revert to the fixed plan) Choose based on the intended nuance.

• danach: “after that,” often referring back to what was just mentioned; slightly more formal/precise than dann in sequences with a clear “after.”
• dann: “then,” general sequencer; very common and versatile.
• nachher: “later/afterwards,” conversational; often near-future or informal.
• später: “later,” more general and can be further in the future.
• anschließend: “subsequently/afterwards,” more formal/written register.

In the sentence, danach fits well because it points to the time immediately after the short-lived solution.