Places in Wirral, especially those mentioned, will appeal but involve up to an hour’s train journey. West Kirby takes some beating; in fact there are so many desirable locations such as Port Sunlight it’s difficult to mention them all.
Short, direct train journeys point to Ainsdale and Birkdale as posters have advised. The frontage at Waterloo/Blundellsands/Hightown/Formby/Hillside (appears as Southport on Rightmove) might lure you.
Car journey: consider Maghull, Lydiate, Lunt, Sefton and Ince Blundell - all quite close to your daughter.
Do check online the area covered by free Merseytravel. Bordering towns and villages can be exempt.